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Practical application of matrices and determinants


Which methods are used by actuaries in practice?Matrix Multiplication - Why Rows $cdot$ Columns = Columns?Why are orthogonal matrices generalizations of rotations and reflections?Question regarding matrices and determinants.The definition of Determinant in the spirit of algebra and geometryApplications of the wave equationSimilarity classes of matricesGeometry Of Unitary TransformationsAdvice for a graduate who is considering studying a second degree in MathsWhat exactly is a matrix?













3












$begingroup$


I have learned recently about matrices and determinants and also about the geometrical interpretations, i.e , how the matrix is used for linear transformations and how determinants tell us about area/volume changes.



My school textbooks tells me that matrices and determinants can be used to solve a system of equations,but I feel that such a vast concept would have more practical applications. My question is: what are the various ways the concept of matrices and determinants is employed in science or everyday life?










share|cite|improve this question









New contributor




Vaishakh Sreekanth Menon is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$











  • $begingroup$
    Matrices are used a lot in machine learning.
    $endgroup$
    – Bladewood
    7 hours ago






  • 2




    $begingroup$
    With some exaggeration, all of applied mathematics boils down to solving systems of linear equations.
    $endgroup$
    – Rodrigo de Azevedo
    7 hours ago






  • 3




    $begingroup$
    Solving systems of equations is extremely practical. Every time someone solves a differential equation using the finite element method, or runs a linear regression, or solves an optimization problem using Newton's method, a system of linear equations is solved. There is hardly any engineering or applied math project that doesn't require solving a system of linear equations.
    $endgroup$
    – Sasho Nikolov
    7 hours ago










  • $begingroup$
    Matrices are important to computer graphics, but not determinants.
    $endgroup$
    – immibis
    4 hours ago















3












$begingroup$


I have learned recently about matrices and determinants and also about the geometrical interpretations, i.e , how the matrix is used for linear transformations and how determinants tell us about area/volume changes.



My school textbooks tells me that matrices and determinants can be used to solve a system of equations,but I feel that such a vast concept would have more practical applications. My question is: what are the various ways the concept of matrices and determinants is employed in science or everyday life?










share|cite|improve this question









New contributor




Vaishakh Sreekanth Menon is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$











  • $begingroup$
    Matrices are used a lot in machine learning.
    $endgroup$
    – Bladewood
    7 hours ago






  • 2




    $begingroup$
    With some exaggeration, all of applied mathematics boils down to solving systems of linear equations.
    $endgroup$
    – Rodrigo de Azevedo
    7 hours ago






  • 3




    $begingroup$
    Solving systems of equations is extremely practical. Every time someone solves a differential equation using the finite element method, or runs a linear regression, or solves an optimization problem using Newton's method, a system of linear equations is solved. There is hardly any engineering or applied math project that doesn't require solving a system of linear equations.
    $endgroup$
    – Sasho Nikolov
    7 hours ago










  • $begingroup$
    Matrices are important to computer graphics, but not determinants.
    $endgroup$
    – immibis
    4 hours ago













3












3








3


1



$begingroup$


I have learned recently about matrices and determinants and also about the geometrical interpretations, i.e , how the matrix is used for linear transformations and how determinants tell us about area/volume changes.



My school textbooks tells me that matrices and determinants can be used to solve a system of equations,but I feel that such a vast concept would have more practical applications. My question is: what are the various ways the concept of matrices and determinants is employed in science or everyday life?










share|cite|improve this question









New contributor




Vaishakh Sreekanth Menon is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$




I have learned recently about matrices and determinants and also about the geometrical interpretations, i.e , how the matrix is used for linear transformations and how determinants tell us about area/volume changes.



My school textbooks tells me that matrices and determinants can be used to solve a system of equations,but I feel that such a vast concept would have more practical applications. My question is: what are the various ways the concept of matrices and determinants is employed in science or everyday life?







matrices soft-question determinant applications






share|cite|improve this question









New contributor




Vaishakh Sreekanth Menon is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question









New contributor




Vaishakh Sreekanth Menon is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









share|cite|improve this question




share|cite|improve this question








edited 11 hours ago









J. W. Tanner

3,3351320




3,3351320






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Vaishakh Sreekanth Menon is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









asked 11 hours ago









Vaishakh Sreekanth MenonVaishakh Sreekanth Menon

192




192




New contributor




Vaishakh Sreekanth Menon is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.





New contributor





Vaishakh Sreekanth Menon is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






Vaishakh Sreekanth Menon is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











  • $begingroup$
    Matrices are used a lot in machine learning.
    $endgroup$
    – Bladewood
    7 hours ago






  • 2




    $begingroup$
    With some exaggeration, all of applied mathematics boils down to solving systems of linear equations.
    $endgroup$
    – Rodrigo de Azevedo
    7 hours ago






  • 3




    $begingroup$
    Solving systems of equations is extremely practical. Every time someone solves a differential equation using the finite element method, or runs a linear regression, or solves an optimization problem using Newton's method, a system of linear equations is solved. There is hardly any engineering or applied math project that doesn't require solving a system of linear equations.
    $endgroup$
    – Sasho Nikolov
    7 hours ago










  • $begingroup$
    Matrices are important to computer graphics, but not determinants.
    $endgroup$
    – immibis
    4 hours ago
















  • $begingroup$
    Matrices are used a lot in machine learning.
    $endgroup$
    – Bladewood
    7 hours ago






  • 2




    $begingroup$
    With some exaggeration, all of applied mathematics boils down to solving systems of linear equations.
    $endgroup$
    – Rodrigo de Azevedo
    7 hours ago






  • 3




    $begingroup$
    Solving systems of equations is extremely practical. Every time someone solves a differential equation using the finite element method, or runs a linear regression, or solves an optimization problem using Newton's method, a system of linear equations is solved. There is hardly any engineering or applied math project that doesn't require solving a system of linear equations.
    $endgroup$
    – Sasho Nikolov
    7 hours ago










  • $begingroup$
    Matrices are important to computer graphics, but not determinants.
    $endgroup$
    – immibis
    4 hours ago















$begingroup$
Matrices are used a lot in machine learning.
$endgroup$
– Bladewood
7 hours ago




$begingroup$
Matrices are used a lot in machine learning.
$endgroup$
– Bladewood
7 hours ago




2




2




$begingroup$
With some exaggeration, all of applied mathematics boils down to solving systems of linear equations.
$endgroup$
– Rodrigo de Azevedo
7 hours ago




$begingroup$
With some exaggeration, all of applied mathematics boils down to solving systems of linear equations.
$endgroup$
– Rodrigo de Azevedo
7 hours ago




3




3




$begingroup$
Solving systems of equations is extremely practical. Every time someone solves a differential equation using the finite element method, or runs a linear regression, or solves an optimization problem using Newton's method, a system of linear equations is solved. There is hardly any engineering or applied math project that doesn't require solving a system of linear equations.
$endgroup$
– Sasho Nikolov
7 hours ago




$begingroup$
Solving systems of equations is extremely practical. Every time someone solves a differential equation using the finite element method, or runs a linear regression, or solves an optimization problem using Newton's method, a system of linear equations is solved. There is hardly any engineering or applied math project that doesn't require solving a system of linear equations.
$endgroup$
– Sasho Nikolov
7 hours ago












$begingroup$
Matrices are important to computer graphics, but not determinants.
$endgroup$
– immibis
4 hours ago




$begingroup$
Matrices are important to computer graphics, but not determinants.
$endgroup$
– immibis
4 hours ago










7 Answers
7






active

oldest

votes


















4












$begingroup$

My first brief understanding of matrices is that they offer an elegant way to deal with data (combinatorially, sort of). A classical and really concrete example would be a discrete Markov chain (don't be frightened by its name). Say you are given the following information: if today is rainy, then tomorrow has a 0.9 probability to be rainy; if today is sunny, then tomorrow has a 0.5 probability to be rainy. Then you may organize these data into a matrix:



$$A=beginpmatrix
0.9 & 0.5 \
0.1 & 0.5
endpmatrix$$



Now if you compute $A^2=beginpmatrix
0.86 & 0.7 \
0.14 & 0.3
endpmatrix$
, what do you get? 0.86 is the probability that if today is rainy then the day after tomorrow is still rainy and 0.7 is the probability that if today is sunny then the day after tomorrow is rainy. And this pattern holds for $A^n$ an arbitrary $n$.



That's the simple point: matrices are a way to calculate elegantly. In my understanding, this aligns with the spirit of mathematics. Math occurs when people try to solve practical problems. People find that if they make good definitions and use good notations, things will be a lot easier. Here comes math. And the matrix is such a good notation to make things easier.






share|cite|improve this answer









$endgroup$




















    3












    $begingroup$

    Matrices are used widely in computer graphics. If you have the coordinates of an object in 3d space, then scaling, stretching and rotating the object can all be done by considering the coordinates to be vectors and multiplying them by the appropriate matrix. When you want to display that object on-screen, the projection down to a 2D object is also a matrix multiplication.






    share|cite|improve this answer









    $endgroup$




















      3












      $begingroup$

      Determinants are of great theoretical significance in mathematics, since in general "the determinant of something $= 0$" means something very special is going on, which may be either good news of bad news depending on the situation.



      On the other hand determinants have very little practical use in numerical calculations, since evaluating a determinant of order $n$ "from first principles" involves $n!$ operations, which is prohibitively expensive unless $n$ is very small. Even Cramer's rule, which is often taught in an introductory course on determinants and matrices, is not the cheapest way to solve $n$ linear equations in $n$ variables numerically if $n>2$, which is a pretty serious limitation!



      Also, if the typical magnitude of each term in a matrix of of order $n$ is $a$, the determinant is likely to be of magnitude $a^n$, and for large $n$ (say $n > 1000$) that number will usually be too large or too small to do efficient computer calculations, unless $|a|$ is very close to $1$.



      On the other hand, almost every type of numerical calculation involves the same techniques that are used to solve equations, so the practical applications of matrices are more or less "the whole of applied mathematics, science, and engineering". Most applications involve systems of equations that are much too big to create and solve by hand, so it is hard to give realistic simple examples. In real-world numerical applications, a set of $n$ linear equations in $n$ variables would still be "small" from a practical point of view if $n = 100,000,$ and even $n = 1,000,000$ is not usually big enough to cause any real problems - the solution would only take a few seconds on a typical personal computer.






      share|cite|improve this answer









      $endgroup$












      • $begingroup$
        Why "even Cramer's rule"? That rule is so obviously inefficient that it's hardly worth mentioning, as every introductory course covers Gaussian elimination, which is clearly much more efficient.
        $endgroup$
        – Servaes
        6 hours ago










      • $begingroup$
        Whilst it doesn't make it more efficient, the determinant calculations in Cramer's rule can be done using Gaussian elimination which means its at least in the same complexity class surely?
        $endgroup$
        – jacob1729
        3 hours ago


















      2












      $begingroup$

      Here's an application in calculus. The multivariate generalisation of integration by substitution viz. $x=f(y)implies dx=f^prime(y)dy$ uses the determinant of a matrix called a Jacobian in place of the $f^prime$ factor. In particular, the chain rule $dx_i=sum_j J_ijdy_j,,J_ij:=fracpartial x_ipartial y_j$ for $n$-dimensional vectors $vecx,,vecy$ can be summarised as $dvecx=Jdvecy$. Then $d^nvecx=|det J|d^nvecy$.






      share|cite|improve this answer









      $endgroup$




















        2












        $begingroup$

        There are plenty of applications of determinants, but I will just mention one that applies to optimization. A totally unimodular matrix is a matrix (doesn’t have to be square) that every square submatrix has a determinant of 0, 1 or -1. It turns out that (by Cramer’s rule) that if a constraint matrix $A$ of a linear program max $c’x:: Ax leq b, x in mathbbR^n_+ $ is totally unimodular, it is guaranteed to have an integer solution if a solution exists. In other words, the polyhedron formed by $P = x:: Ax leq b$ has integer vertices in $mathbbR^n$. This has major implications in integer programming, as we solve an integer program that has a totally unimodular matrix as a linear program. This is advantageous because a linear program can me solved in polynomial time, where there is no polynomial algorithm for integer programs.






        share|cite|improve this answer









        $endgroup$




















          1












          $begingroup$

          If the determinant of a matrix is zero, then there are no solutions to a set of equations represented by an nXn matrix set equal to a 1Xn matrix. If it is non-zero, then there are solutions and they can all be found using Cramer's Rule. They are also used in Photoshop for various visual tricks; they are used to cast 3D shapes onto a 2D surface; they are used to analyze seismic waves... and a hundred other applications where data need to be crunched in a simple manner.






          share|cite|improve this answer









          $endgroup$




















            1












            $begingroup$

            Besides the applications already mentioned in the previous answers, just consider that matrices are the fundamental basis for Finite Elements design, today widely used in any sector of engineering.

            Also, in the continuous analysis of the deformation of bodies, stress and strain each are represented by matrices (tensors).

            And the inertia of a body to rotation is a matrix






            share|cite|improve this answer









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              7 Answers
              7






              active

              oldest

              votes








              7 Answers
              7






              active

              oldest

              votes









              active

              oldest

              votes






              active

              oldest

              votes









              4












              $begingroup$

              My first brief understanding of matrices is that they offer an elegant way to deal with data (combinatorially, sort of). A classical and really concrete example would be a discrete Markov chain (don't be frightened by its name). Say you are given the following information: if today is rainy, then tomorrow has a 0.9 probability to be rainy; if today is sunny, then tomorrow has a 0.5 probability to be rainy. Then you may organize these data into a matrix:



              $$A=beginpmatrix
              0.9 & 0.5 \
              0.1 & 0.5
              endpmatrix$$



              Now if you compute $A^2=beginpmatrix
              0.86 & 0.7 \
              0.14 & 0.3
              endpmatrix$
              , what do you get? 0.86 is the probability that if today is rainy then the day after tomorrow is still rainy and 0.7 is the probability that if today is sunny then the day after tomorrow is rainy. And this pattern holds for $A^n$ an arbitrary $n$.



              That's the simple point: matrices are a way to calculate elegantly. In my understanding, this aligns with the spirit of mathematics. Math occurs when people try to solve practical problems. People find that if they make good definitions and use good notations, things will be a lot easier. Here comes math. And the matrix is such a good notation to make things easier.






              share|cite|improve this answer









              $endgroup$

















                4












                $begingroup$

                My first brief understanding of matrices is that they offer an elegant way to deal with data (combinatorially, sort of). A classical and really concrete example would be a discrete Markov chain (don't be frightened by its name). Say you are given the following information: if today is rainy, then tomorrow has a 0.9 probability to be rainy; if today is sunny, then tomorrow has a 0.5 probability to be rainy. Then you may organize these data into a matrix:



                $$A=beginpmatrix
                0.9 & 0.5 \
                0.1 & 0.5
                endpmatrix$$



                Now if you compute $A^2=beginpmatrix
                0.86 & 0.7 \
                0.14 & 0.3
                endpmatrix$
                , what do you get? 0.86 is the probability that if today is rainy then the day after tomorrow is still rainy and 0.7 is the probability that if today is sunny then the day after tomorrow is rainy. And this pattern holds for $A^n$ an arbitrary $n$.



                That's the simple point: matrices are a way to calculate elegantly. In my understanding, this aligns with the spirit of mathematics. Math occurs when people try to solve practical problems. People find that if they make good definitions and use good notations, things will be a lot easier. Here comes math. And the matrix is such a good notation to make things easier.






                share|cite|improve this answer









                $endgroup$















                  4












                  4








                  4





                  $begingroup$

                  My first brief understanding of matrices is that they offer an elegant way to deal with data (combinatorially, sort of). A classical and really concrete example would be a discrete Markov chain (don't be frightened by its name). Say you are given the following information: if today is rainy, then tomorrow has a 0.9 probability to be rainy; if today is sunny, then tomorrow has a 0.5 probability to be rainy. Then you may organize these data into a matrix:



                  $$A=beginpmatrix
                  0.9 & 0.5 \
                  0.1 & 0.5
                  endpmatrix$$



                  Now if you compute $A^2=beginpmatrix
                  0.86 & 0.7 \
                  0.14 & 0.3
                  endpmatrix$
                  , what do you get? 0.86 is the probability that if today is rainy then the day after tomorrow is still rainy and 0.7 is the probability that if today is sunny then the day after tomorrow is rainy. And this pattern holds for $A^n$ an arbitrary $n$.



                  That's the simple point: matrices are a way to calculate elegantly. In my understanding, this aligns with the spirit of mathematics. Math occurs when people try to solve practical problems. People find that if they make good definitions and use good notations, things will be a lot easier. Here comes math. And the matrix is such a good notation to make things easier.






                  share|cite|improve this answer









                  $endgroup$



                  My first brief understanding of matrices is that they offer an elegant way to deal with data (combinatorially, sort of). A classical and really concrete example would be a discrete Markov chain (don't be frightened by its name). Say you are given the following information: if today is rainy, then tomorrow has a 0.9 probability to be rainy; if today is sunny, then tomorrow has a 0.5 probability to be rainy. Then you may organize these data into a matrix:



                  $$A=beginpmatrix
                  0.9 & 0.5 \
                  0.1 & 0.5
                  endpmatrix$$



                  Now if you compute $A^2=beginpmatrix
                  0.86 & 0.7 \
                  0.14 & 0.3
                  endpmatrix$
                  , what do you get? 0.86 is the probability that if today is rainy then the day after tomorrow is still rainy and 0.7 is the probability that if today is sunny then the day after tomorrow is rainy. And this pattern holds for $A^n$ an arbitrary $n$.



                  That's the simple point: matrices are a way to calculate elegantly. In my understanding, this aligns with the spirit of mathematics. Math occurs when people try to solve practical problems. People find that if they make good definitions and use good notations, things will be a lot easier. Here comes math. And the matrix is such a good notation to make things easier.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 10 hours ago









                  J. WangJ. Wang

                  945




                  945





















                      3












                      $begingroup$

                      Matrices are used widely in computer graphics. If you have the coordinates of an object in 3d space, then scaling, stretching and rotating the object can all be done by considering the coordinates to be vectors and multiplying them by the appropriate matrix. When you want to display that object on-screen, the projection down to a 2D object is also a matrix multiplication.






                      share|cite|improve this answer









                      $endgroup$

















                        3












                        $begingroup$

                        Matrices are used widely in computer graphics. If you have the coordinates of an object in 3d space, then scaling, stretching and rotating the object can all be done by considering the coordinates to be vectors and multiplying them by the appropriate matrix. When you want to display that object on-screen, the projection down to a 2D object is also a matrix multiplication.






                        share|cite|improve this answer









                        $endgroup$















                          3












                          3








                          3





                          $begingroup$

                          Matrices are used widely in computer graphics. If you have the coordinates of an object in 3d space, then scaling, stretching and rotating the object can all be done by considering the coordinates to be vectors and multiplying them by the appropriate matrix. When you want to display that object on-screen, the projection down to a 2D object is also a matrix multiplication.






                          share|cite|improve this answer









                          $endgroup$



                          Matrices are used widely in computer graphics. If you have the coordinates of an object in 3d space, then scaling, stretching and rotating the object can all be done by considering the coordinates to be vectors and multiplying them by the appropriate matrix. When you want to display that object on-screen, the projection down to a 2D object is also a matrix multiplication.







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered 7 hours ago









                          David RicherbyDavid Richerby

                          2,22511324




                          2,22511324





















                              3












                              $begingroup$

                              Determinants are of great theoretical significance in mathematics, since in general "the determinant of something $= 0$" means something very special is going on, which may be either good news of bad news depending on the situation.



                              On the other hand determinants have very little practical use in numerical calculations, since evaluating a determinant of order $n$ "from first principles" involves $n!$ operations, which is prohibitively expensive unless $n$ is very small. Even Cramer's rule, which is often taught in an introductory course on determinants and matrices, is not the cheapest way to solve $n$ linear equations in $n$ variables numerically if $n>2$, which is a pretty serious limitation!



                              Also, if the typical magnitude of each term in a matrix of of order $n$ is $a$, the determinant is likely to be of magnitude $a^n$, and for large $n$ (say $n > 1000$) that number will usually be too large or too small to do efficient computer calculations, unless $|a|$ is very close to $1$.



                              On the other hand, almost every type of numerical calculation involves the same techniques that are used to solve equations, so the practical applications of matrices are more or less "the whole of applied mathematics, science, and engineering". Most applications involve systems of equations that are much too big to create and solve by hand, so it is hard to give realistic simple examples. In real-world numerical applications, a set of $n$ linear equations in $n$ variables would still be "small" from a practical point of view if $n = 100,000,$ and even $n = 1,000,000$ is not usually big enough to cause any real problems - the solution would only take a few seconds on a typical personal computer.






                              share|cite|improve this answer









                              $endgroup$












                              • $begingroup$
                                Why "even Cramer's rule"? That rule is so obviously inefficient that it's hardly worth mentioning, as every introductory course covers Gaussian elimination, which is clearly much more efficient.
                                $endgroup$
                                – Servaes
                                6 hours ago










                              • $begingroup$
                                Whilst it doesn't make it more efficient, the determinant calculations in Cramer's rule can be done using Gaussian elimination which means its at least in the same complexity class surely?
                                $endgroup$
                                – jacob1729
                                3 hours ago















                              3












                              $begingroup$

                              Determinants are of great theoretical significance in mathematics, since in general "the determinant of something $= 0$" means something very special is going on, which may be either good news of bad news depending on the situation.



                              On the other hand determinants have very little practical use in numerical calculations, since evaluating a determinant of order $n$ "from first principles" involves $n!$ operations, which is prohibitively expensive unless $n$ is very small. Even Cramer's rule, which is often taught in an introductory course on determinants and matrices, is not the cheapest way to solve $n$ linear equations in $n$ variables numerically if $n>2$, which is a pretty serious limitation!



                              Also, if the typical magnitude of each term in a matrix of of order $n$ is $a$, the determinant is likely to be of magnitude $a^n$, and for large $n$ (say $n > 1000$) that number will usually be too large or too small to do efficient computer calculations, unless $|a|$ is very close to $1$.



                              On the other hand, almost every type of numerical calculation involves the same techniques that are used to solve equations, so the practical applications of matrices are more or less "the whole of applied mathematics, science, and engineering". Most applications involve systems of equations that are much too big to create and solve by hand, so it is hard to give realistic simple examples. In real-world numerical applications, a set of $n$ linear equations in $n$ variables would still be "small" from a practical point of view if $n = 100,000,$ and even $n = 1,000,000$ is not usually big enough to cause any real problems - the solution would only take a few seconds on a typical personal computer.






                              share|cite|improve this answer









                              $endgroup$












                              • $begingroup$
                                Why "even Cramer's rule"? That rule is so obviously inefficient that it's hardly worth mentioning, as every introductory course covers Gaussian elimination, which is clearly much more efficient.
                                $endgroup$
                                – Servaes
                                6 hours ago










                              • $begingroup$
                                Whilst it doesn't make it more efficient, the determinant calculations in Cramer's rule can be done using Gaussian elimination which means its at least in the same complexity class surely?
                                $endgroup$
                                – jacob1729
                                3 hours ago













                              3












                              3








                              3





                              $begingroup$

                              Determinants are of great theoretical significance in mathematics, since in general "the determinant of something $= 0$" means something very special is going on, which may be either good news of bad news depending on the situation.



                              On the other hand determinants have very little practical use in numerical calculations, since evaluating a determinant of order $n$ "from first principles" involves $n!$ operations, which is prohibitively expensive unless $n$ is very small. Even Cramer's rule, which is often taught in an introductory course on determinants and matrices, is not the cheapest way to solve $n$ linear equations in $n$ variables numerically if $n>2$, which is a pretty serious limitation!



                              Also, if the typical magnitude of each term in a matrix of of order $n$ is $a$, the determinant is likely to be of magnitude $a^n$, and for large $n$ (say $n > 1000$) that number will usually be too large or too small to do efficient computer calculations, unless $|a|$ is very close to $1$.



                              On the other hand, almost every type of numerical calculation involves the same techniques that are used to solve equations, so the practical applications of matrices are more or less "the whole of applied mathematics, science, and engineering". Most applications involve systems of equations that are much too big to create and solve by hand, so it is hard to give realistic simple examples. In real-world numerical applications, a set of $n$ linear equations in $n$ variables would still be "small" from a practical point of view if $n = 100,000,$ and even $n = 1,000,000$ is not usually big enough to cause any real problems - the solution would only take a few seconds on a typical personal computer.






                              share|cite|improve this answer









                              $endgroup$



                              Determinants are of great theoretical significance in mathematics, since in general "the determinant of something $= 0$" means something very special is going on, which may be either good news of bad news depending on the situation.



                              On the other hand determinants have very little practical use in numerical calculations, since evaluating a determinant of order $n$ "from first principles" involves $n!$ operations, which is prohibitively expensive unless $n$ is very small. Even Cramer's rule, which is often taught in an introductory course on determinants and matrices, is not the cheapest way to solve $n$ linear equations in $n$ variables numerically if $n>2$, which is a pretty serious limitation!



                              Also, if the typical magnitude of each term in a matrix of of order $n$ is $a$, the determinant is likely to be of magnitude $a^n$, and for large $n$ (say $n > 1000$) that number will usually be too large or too small to do efficient computer calculations, unless $|a|$ is very close to $1$.



                              On the other hand, almost every type of numerical calculation involves the same techniques that are used to solve equations, so the practical applications of matrices are more or less "the whole of applied mathematics, science, and engineering". Most applications involve systems of equations that are much too big to create and solve by hand, so it is hard to give realistic simple examples. In real-world numerical applications, a set of $n$ linear equations in $n$ variables would still be "small" from a practical point of view if $n = 100,000,$ and even $n = 1,000,000$ is not usually big enough to cause any real problems - the solution would only take a few seconds on a typical personal computer.







                              share|cite|improve this answer












                              share|cite|improve this answer



                              share|cite|improve this answer










                              answered 7 hours ago









                              alephzeroalephzero

                              66037




                              66037











                              • $begingroup$
                                Why "even Cramer's rule"? That rule is so obviously inefficient that it's hardly worth mentioning, as every introductory course covers Gaussian elimination, which is clearly much more efficient.
                                $endgroup$
                                – Servaes
                                6 hours ago










                              • $begingroup$
                                Whilst it doesn't make it more efficient, the determinant calculations in Cramer's rule can be done using Gaussian elimination which means its at least in the same complexity class surely?
                                $endgroup$
                                – jacob1729
                                3 hours ago
















                              • $begingroup$
                                Why "even Cramer's rule"? That rule is so obviously inefficient that it's hardly worth mentioning, as every introductory course covers Gaussian elimination, which is clearly much more efficient.
                                $endgroup$
                                – Servaes
                                6 hours ago










                              • $begingroup$
                                Whilst it doesn't make it more efficient, the determinant calculations in Cramer's rule can be done using Gaussian elimination which means its at least in the same complexity class surely?
                                $endgroup$
                                – jacob1729
                                3 hours ago















                              $begingroup$
                              Why "even Cramer's rule"? That rule is so obviously inefficient that it's hardly worth mentioning, as every introductory course covers Gaussian elimination, which is clearly much more efficient.
                              $endgroup$
                              – Servaes
                              6 hours ago




                              $begingroup$
                              Why "even Cramer's rule"? That rule is so obviously inefficient that it's hardly worth mentioning, as every introductory course covers Gaussian elimination, which is clearly much more efficient.
                              $endgroup$
                              – Servaes
                              6 hours ago












                              $begingroup$
                              Whilst it doesn't make it more efficient, the determinant calculations in Cramer's rule can be done using Gaussian elimination which means its at least in the same complexity class surely?
                              $endgroup$
                              – jacob1729
                              3 hours ago




                              $begingroup$
                              Whilst it doesn't make it more efficient, the determinant calculations in Cramer's rule can be done using Gaussian elimination which means its at least in the same complexity class surely?
                              $endgroup$
                              – jacob1729
                              3 hours ago











                              2












                              $begingroup$

                              Here's an application in calculus. The multivariate generalisation of integration by substitution viz. $x=f(y)implies dx=f^prime(y)dy$ uses the determinant of a matrix called a Jacobian in place of the $f^prime$ factor. In particular, the chain rule $dx_i=sum_j J_ijdy_j,,J_ij:=fracpartial x_ipartial y_j$ for $n$-dimensional vectors $vecx,,vecy$ can be summarised as $dvecx=Jdvecy$. Then $d^nvecx=|det J|d^nvecy$.






                              share|cite|improve this answer









                              $endgroup$

















                                2












                                $begingroup$

                                Here's an application in calculus. The multivariate generalisation of integration by substitution viz. $x=f(y)implies dx=f^prime(y)dy$ uses the determinant of a matrix called a Jacobian in place of the $f^prime$ factor. In particular, the chain rule $dx_i=sum_j J_ijdy_j,,J_ij:=fracpartial x_ipartial y_j$ for $n$-dimensional vectors $vecx,,vecy$ can be summarised as $dvecx=Jdvecy$. Then $d^nvecx=|det J|d^nvecy$.






                                share|cite|improve this answer









                                $endgroup$















                                  2












                                  2








                                  2





                                  $begingroup$

                                  Here's an application in calculus. The multivariate generalisation of integration by substitution viz. $x=f(y)implies dx=f^prime(y)dy$ uses the determinant of a matrix called a Jacobian in place of the $f^prime$ factor. In particular, the chain rule $dx_i=sum_j J_ijdy_j,,J_ij:=fracpartial x_ipartial y_j$ for $n$-dimensional vectors $vecx,,vecy$ can be summarised as $dvecx=Jdvecy$. Then $d^nvecx=|det J|d^nvecy$.






                                  share|cite|improve this answer









                                  $endgroup$



                                  Here's an application in calculus. The multivariate generalisation of integration by substitution viz. $x=f(y)implies dx=f^prime(y)dy$ uses the determinant of a matrix called a Jacobian in place of the $f^prime$ factor. In particular, the chain rule $dx_i=sum_j J_ijdy_j,,J_ij:=fracpartial x_ipartial y_j$ for $n$-dimensional vectors $vecx,,vecy$ can be summarised as $dvecx=Jdvecy$. Then $d^nvecx=|det J|d^nvecy$.







                                  share|cite|improve this answer












                                  share|cite|improve this answer



                                  share|cite|improve this answer










                                  answered 11 hours ago









                                  J.G.J.G.

                                  30.5k23148




                                  30.5k23148





















                                      2












                                      $begingroup$

                                      There are plenty of applications of determinants, but I will just mention one that applies to optimization. A totally unimodular matrix is a matrix (doesn’t have to be square) that every square submatrix has a determinant of 0, 1 or -1. It turns out that (by Cramer’s rule) that if a constraint matrix $A$ of a linear program max $c’x:: Ax leq b, x in mathbbR^n_+ $ is totally unimodular, it is guaranteed to have an integer solution if a solution exists. In other words, the polyhedron formed by $P = x:: Ax leq b$ has integer vertices in $mathbbR^n$. This has major implications in integer programming, as we solve an integer program that has a totally unimodular matrix as a linear program. This is advantageous because a linear program can me solved in polynomial time, where there is no polynomial algorithm for integer programs.






                                      share|cite|improve this answer









                                      $endgroup$

















                                        2












                                        $begingroup$

                                        There are plenty of applications of determinants, but I will just mention one that applies to optimization. A totally unimodular matrix is a matrix (doesn’t have to be square) that every square submatrix has a determinant of 0, 1 or -1. It turns out that (by Cramer’s rule) that if a constraint matrix $A$ of a linear program max $c’x:: Ax leq b, x in mathbbR^n_+ $ is totally unimodular, it is guaranteed to have an integer solution if a solution exists. In other words, the polyhedron formed by $P = x:: Ax leq b$ has integer vertices in $mathbbR^n$. This has major implications in integer programming, as we solve an integer program that has a totally unimodular matrix as a linear program. This is advantageous because a linear program can me solved in polynomial time, where there is no polynomial algorithm for integer programs.






                                        share|cite|improve this answer









                                        $endgroup$















                                          2












                                          2








                                          2





                                          $begingroup$

                                          There are plenty of applications of determinants, but I will just mention one that applies to optimization. A totally unimodular matrix is a matrix (doesn’t have to be square) that every square submatrix has a determinant of 0, 1 or -1. It turns out that (by Cramer’s rule) that if a constraint matrix $A$ of a linear program max $c’x:: Ax leq b, x in mathbbR^n_+ $ is totally unimodular, it is guaranteed to have an integer solution if a solution exists. In other words, the polyhedron formed by $P = x:: Ax leq b$ has integer vertices in $mathbbR^n$. This has major implications in integer programming, as we solve an integer program that has a totally unimodular matrix as a linear program. This is advantageous because a linear program can me solved in polynomial time, where there is no polynomial algorithm for integer programs.






                                          share|cite|improve this answer









                                          $endgroup$



                                          There are plenty of applications of determinants, but I will just mention one that applies to optimization. A totally unimodular matrix is a matrix (doesn’t have to be square) that every square submatrix has a determinant of 0, 1 or -1. It turns out that (by Cramer’s rule) that if a constraint matrix $A$ of a linear program max $c’x:: Ax leq b, x in mathbbR^n_+ $ is totally unimodular, it is guaranteed to have an integer solution if a solution exists. In other words, the polyhedron formed by $P = x:: Ax leq b$ has integer vertices in $mathbbR^n$. This has major implications in integer programming, as we solve an integer program that has a totally unimodular matrix as a linear program. This is advantageous because a linear program can me solved in polynomial time, where there is no polynomial algorithm for integer programs.







                                          share|cite|improve this answer












                                          share|cite|improve this answer



                                          share|cite|improve this answer










                                          answered 11 hours ago









                                          JBLJBL

                                          473210




                                          473210





















                                              1












                                              $begingroup$

                                              If the determinant of a matrix is zero, then there are no solutions to a set of equations represented by an nXn matrix set equal to a 1Xn matrix. If it is non-zero, then there are solutions and they can all be found using Cramer's Rule. They are also used in Photoshop for various visual tricks; they are used to cast 3D shapes onto a 2D surface; they are used to analyze seismic waves... and a hundred other applications where data need to be crunched in a simple manner.






                                              share|cite|improve this answer









                                              $endgroup$

















                                                1












                                                $begingroup$

                                                If the determinant of a matrix is zero, then there are no solutions to a set of equations represented by an nXn matrix set equal to a 1Xn matrix. If it is non-zero, then there are solutions and they can all be found using Cramer's Rule. They are also used in Photoshop for various visual tricks; they are used to cast 3D shapes onto a 2D surface; they are used to analyze seismic waves... and a hundred other applications where data need to be crunched in a simple manner.






                                                share|cite|improve this answer









                                                $endgroup$















                                                  1












                                                  1








                                                  1





                                                  $begingroup$

                                                  If the determinant of a matrix is zero, then there are no solutions to a set of equations represented by an nXn matrix set equal to a 1Xn matrix. If it is non-zero, then there are solutions and they can all be found using Cramer's Rule. They are also used in Photoshop for various visual tricks; they are used to cast 3D shapes onto a 2D surface; they are used to analyze seismic waves... and a hundred other applications where data need to be crunched in a simple manner.






                                                  share|cite|improve this answer









                                                  $endgroup$



                                                  If the determinant of a matrix is zero, then there are no solutions to a set of equations represented by an nXn matrix set equal to a 1Xn matrix. If it is non-zero, then there are solutions and they can all be found using Cramer's Rule. They are also used in Photoshop for various visual tricks; they are used to cast 3D shapes onto a 2D surface; they are used to analyze seismic waves... and a hundred other applications where data need to be crunched in a simple manner.







                                                  share|cite|improve this answer












                                                  share|cite|improve this answer



                                                  share|cite|improve this answer










                                                  answered 11 hours ago









                                                  poetasispoetasis

                                                  400217




                                                  400217





















                                                      1












                                                      $begingroup$

                                                      Besides the applications already mentioned in the previous answers, just consider that matrices are the fundamental basis for Finite Elements design, today widely used in any sector of engineering.

                                                      Also, in the continuous analysis of the deformation of bodies, stress and strain each are represented by matrices (tensors).

                                                      And the inertia of a body to rotation is a matrix






                                                      share|cite|improve this answer









                                                      $endgroup$

















                                                        1












                                                        $begingroup$

                                                        Besides the applications already mentioned in the previous answers, just consider that matrices are the fundamental basis for Finite Elements design, today widely used in any sector of engineering.

                                                        Also, in the continuous analysis of the deformation of bodies, stress and strain each are represented by matrices (tensors).

                                                        And the inertia of a body to rotation is a matrix






                                                        share|cite|improve this answer









                                                        $endgroup$















                                                          1












                                                          1








                                                          1





                                                          $begingroup$

                                                          Besides the applications already mentioned in the previous answers, just consider that matrices are the fundamental basis for Finite Elements design, today widely used in any sector of engineering.

                                                          Also, in the continuous analysis of the deformation of bodies, stress and strain each are represented by matrices (tensors).

                                                          And the inertia of a body to rotation is a matrix






                                                          share|cite|improve this answer









                                                          $endgroup$



                                                          Besides the applications already mentioned in the previous answers, just consider that matrices are the fundamental basis for Finite Elements design, today widely used in any sector of engineering.

                                                          Also, in the continuous analysis of the deformation of bodies, stress and strain each are represented by matrices (tensors).

                                                          And the inertia of a body to rotation is a matrix







                                                          share|cite|improve this answer












                                                          share|cite|improve this answer



                                                          share|cite|improve this answer










                                                          answered 6 hours ago









                                                          G CabG Cab

                                                          20.1k31340




                                                          20.1k31340




















                                                              Vaishakh Sreekanth Menon is a new contributor. Be nice, and check out our Code of Conduct.









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                                                              Vaishakh Sreekanth Menon is a new contributor. Be nice, and check out our Code of Conduct.












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