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Product of Matrices

This study material provides detailed information about the multiplication, application, and operation of matrices.

Table of Content

A matrix is a form of representation of numbers or equations arranged in rows and columns like a rectangular table or box. The number of rows and columns specify the dimensions of the matrix. For example, if the matrix has 3 rows and 4 columns, it is said to be a 3*4 matrix. The product of matrices follows certain rules and linear algebra related to matrices. Matrix multiplication is based on arithmetic operations and has much application in the mathematics and research domain. It provides a base for transforming and analysing complex values and relations stored in a matrix. 

Representation of a Matrix

The initial step is to write a matrix in the proper format so that arithmetic operations can be performed easily.

Suppose a matrix X with 3 rows and 4 columns.

It is represented as Xmn, where m and n represent the number of rows and columns, respectively. Therefore, Xmn = X34

Let’s fill the matrix X with some values, where, X3*4 = {"aid":null,"font":{"family":"Calibri","size":12,"color":"#0e101a"},"id":"3","code":"$$\\begin{bmatrix}\n{5}&{8}&{9}&{7}\\\\\n{4}&{2}&{0}&{4}\\\\\n{4}&{6}&{3}&{0}\\\\\n\\end{bmatrix}$$","type":"$$","backgroundColor":"#ffffff","backgroundColorModified":null,"ts":1647065527167,"cs":"PGHh59pNk5i4DYvGApnE4A==","size":{"width":92,"height":60}}

 

Here, the space indicates a new column has started, and a comma, i.e., ‘,’ represents a new row.

For example, 5 belongs to the first row and the first column, and 3 belongs to the third row and third column.

Properties of the Product of Matrices

There are various basic properties associated with the product of matrices that include: 

 

  • Commutative Property: The product of two matrices, i.e., X and Y, is not commutative. Hence, XY ≠ YX.

  • Associative Property: The product of three matrices, i.e., X, Y, and Z, is associative. Therefore, X(YZ) = (XY)Z.

  • Distributive Property: The product of three matrices, i.e., X, Y, and Z, is distributive. Therefore, X(Y + Z) = XY + YZ.

Product of Matrices – Matrix Multiplied by Scalar

The multiplication of a matrix by a scalar quantity is the basic multiplication method. It is the simplest and fastest multiplication of a matrix where all the values of the matrix are multiplied by the scalar value.

 

Let’s take a matrix X3*2 = {"aid":null,"type":"$$","id":"4","backgroundColor":"#ffffff","backgroundColorModified":false,"code":"$$\\begin{bmatrix}\n{1}&{2}\\\\\n{8}&{0}\\\\\n{9}&{3}\\\\\n\\end{bmatrix}$$","font":{"family":"Calibri","size":12.5,"color":"#0e101a"},"ts":1647065672461,"cs":"4jtvBHrlKva7DhTTbjsplw==","size":{"width":46,"height":65}} and a scalar value 5.

 

The product of matrix X and scalar value 5 is given as 

 

P = 5*X3*2 = 5 {"aid":null,"type":"$$","id":"4","backgroundColor":"#ffffff","backgroundColorModified":false,"code":"$$\\begin{bmatrix}\n{1}&{2}\\\\\n{8}&{0}\\\\\n{9}&{3}\\\\\n\\end{bmatrix}$$","font":{"family":"Calibri","size":12.5,"color":"#0e101a"},"ts":1647065672461,"cs":"4jtvBHrlKva7DhTTbjsplw==","size":{"width":46,"height":65}}



The final product is P = {"aid":null,"font":{"color":"#0e101a","size":12,"family":"Calibri"},"backgroundColorModified":false,"backgroundColor":"#ffffff","code":"$$\\begin{bmatrix}\n{5}&{12}\\\\\n{40}&{0}\\\\\n{45}&{15}\\\\\n\\end{bmatrix}$$","id":"5","type":"$$","ts":1647065761124,"cs":"g1sWgpNWPzqphVYAgiPGpQ==","size":{"width":60,"height":60}}

 

All the row and column values get multiplied by the scalar value. It does not change the dimension of the matrix.

Conditions for the Product of Matrices

Matrix multiplication is possible only if the number of columns in the first matrix is equal to the number of rows in the second matrix. This ensures that each value of both matrices gets multiplied and forms an element in the new matrix.

 

Let’s take two matrices X3*3 and Y3*2 where the number of columns of matrix X is equal to the number of rows of matrix Y, i.e., both have a value = 3.

 

The matrix product is represented by P = XY.

 

In the resulting matrix, the number of rows and columns is equal to the number of rows of the first matrix, i.e., X, and the number of columns of the second matrix, i.e., Y, respectively.

 

Here, the resulting matrix P is represented by P3*2.

 

In general, if X and Y have dimensions (m,n) and (q,r) respectively, represented as Xm*nand Yq*r, the P, i.e., the matrix formed by the product of X and Y, is represented as Pm*r.

 

The dimension of the resultant matrix changes in this case, i.e., it is not equal to the dimension of any of the parent matrices.

Examples

Let’s take two matrices, X2*3 = {"type":"$$","aid":null,"id":"6","backgroundColor":"#ffffff","code":"$$\\begin{bmatrix}\n{2}&{1}&{0}\\\\\n{3}&{0}&{4}\\\\\n\\end{bmatrix}$$","backgroundColorModified":false,"font":{"color":"#0e101a","size":12,"family":"Calibri"},"ts":1647065835475,"cs":"dUQD40q57F9zKTJlqOWTFg==","size":{"width":65,"height":38}} and Y3*4 = {"code":"$$\\begin{bmatrix}\n{4}&{0}&{1}&{2}\\\\\n{4}&{8}&{9}&{7}\\\\\n{5}&{2}&{3}&{6}\\\\\n\\end{bmatrix}$$","id":"7","type":"$$","backgroundColorModified":false,"aid":null,"backgroundColor":"#ffffff","font":{"color":"#0e101a","size":12,"family":"Calibri"},"ts":1647065899765,"cs":"ZXd5pUDkCZW9ag5FjsSNFg==","size":{"width":92,"height":60}}

 

Here, the number of rows of matrix X is equal to the number of columns of matrix Y, i.e., both are equal to 3.

 

Now, multiply the matrices X and Y:

 

X*Y = {"type":"$$","aid":null,"id":"6","backgroundColor":"#ffffff","code":"$$\\begin{bmatrix}\n{2}&{1}&{0}\\\\\n{3}&{0}&{4}\\\\\n\\end{bmatrix}$$","backgroundColorModified":false,"font":{"color":"#0e101a","size":12,"family":"Calibri"},"ts":1647065835475,"cs":"dUQD40q57F9zKTJlqOWTFg==","size":{"width":65,"height":38}}* {"code":"$$\\begin{bmatrix}\n{4}&{0}&{1}&{2}\\\\\n{4}&{8}&{9}&{7}\\\\\n{5}&{2}&{3}&{6}\\\\\n\\end{bmatrix}$$","id":"7","type":"$$","backgroundColorModified":false,"aid":null,"backgroundColor":"#ffffff","font":{"color":"#0e101a","size":12,"family":"Calibri"},"ts":1647065899765,"cs":"ZXd5pUDkCZW9ag5FjsSNFg==","size":{"width":92,"height":60}}

 

Here, multiply the first row of matrix X with all the columns of matrix Y one at a time and iterate the process for the remaining rows of matrix X.

 

Therefore,

 

{"id":"9","font":{"size":12,"color":"#0e101a","family":"Calibri"},"backgroundColorModified":null,"backgroundColor":"#ffffff","aid":null,"code":"$\\begin{bmatrix}\n{2}&{1}&{0}\\\\\n\\end{bmatrix}$","type":"$","ts":1647067457444,"cs":"V2pCCjMRc1Wj+eBHM1kccA==","size":{"width":61,"height":16}}* {"code":"$$\\begin{bmatrix}\n{4}\\\\\n{4}\\\\\n{5}\\\\\n\\end{bmatrix}$$","backgroundColorModified":false,"backgroundColor":"#ffffff","type":"$$","font":{"color":"#0e101a","size":12,"family":"Calibri"},"aid":null,"id":"10","ts":1647067489417,"cs":"ZfkkT4k+eAT7z5Xo9sZgvQ==","size":{"width":18,"height":60}} = 2*4 + 1*4 + 0*5 = 12

 

  {"id":"9","font":{"size":12,"color":"#0e101a","family":"Calibri"},"backgroundColorModified":null,"backgroundColor":"#ffffff","aid":null,"code":"$\\begin{bmatrix}\n{2}&{1}&{0}\\\\\n\\end{bmatrix}$","type":"$","ts":1647067457444,"cs":"V2pCCjMRc1Wj+eBHM1kccA==","size":{"width":61,"height":16}}* {"font":{"size":12,"color":"#0e101a","family":"Calibri"},"code":"$$\\begin{bmatrix}\n{0}\\\\\n{8}\\\\\n{2}\\\\\n\\end{bmatrix}$$","aid":null,"backgroundColorModified":false,"type":"$$","backgroundColor":"#ffffff","id":"11","ts":1647067531571,"cs":"mAClrlaJyi4MMl2OSdweEw==","size":{"width":18,"height":60}} = 8

 

{"id":"9","font":{"size":12,"color":"#0e101a","family":"Calibri"},"backgroundColorModified":null,"backgroundColor":"#ffffff","aid":null,"code":"$\\begin{bmatrix}\n{2}&{1}&{0}\\\\\n\\end{bmatrix}$","type":"$","ts":1647067457444,"cs":"V2pCCjMRc1Wj+eBHM1kccA==","size":{"width":61,"height":16}}* {"font":{"size":12,"family":"Calibri","color":"#0e101a"},"id":"12","type":"$$","backgroundColor":"#ffffff","aid":null,"code":"$$\\begin{bmatrix}\n{1}\\\\\n{9}\\\\\n{3}\\\\\n\\end{bmatrix}$$","backgroundColorModified":false,"ts":1647067580205,"cs":"BmPU0/ACuOPyyJod54Q/9g==","size":{"width":18,"height":60}} = 11

 

{"id":"9","font":{"size":12,"color":"#0e101a","family":"Calibri"},"backgroundColorModified":null,"backgroundColor":"#ffffff","aid":null,"code":"$\\begin{bmatrix}\n{2}&{1}&{0}\\\\\n\\end{bmatrix}$","type":"$","ts":1647067457444,"cs":"V2pCCjMRc1Wj+eBHM1kccA==","size":{"width":61,"height":16}}* {"font":{"color":"#0e101a","family":"Calibri","size":12},"type":"$$","backgroundColorModified":false,"aid":null,"code":"$$\\begin{bmatrix}\n{2}\\\\\n{7}\\\\\n{6}\\\\\n\\end{bmatrix}$$","backgroundColor":"#ffffff","id":"13","ts":1647067608402,"cs":"VCmohIyosxF/586rat0qjg==","size":{"width":18,"height":60}} = 11

 

Similarly, for the second row of matrix X, 

 

{"font":{"family":"Calibri","size":12,"color":"#0e101a"},"type":"$$","aid":null,"backgroundColor":"#ffffff","code":"$$\\begin{bmatrix}\n{3}&{0}&{4}\\\\\n\\end{bmatrix}$$","id":"14","backgroundColorModified":false,"ts":1647067650358,"cs":"7rK9+jQS9ehSeRLGu5eySA==","size":{"width":61,"height":16}}* {"code":"$$\\begin{bmatrix}\n{4}\\\\\n{4}\\\\\n{5}\\\\\n\\end{bmatrix}$$","backgroundColorModified":false,"backgroundColor":"#ffffff","type":"$$","font":{"color":"#0e101a","size":12,"family":"Calibri"},"aid":null,"id":"10","ts":1647067489417,"cs":"ZfkkT4k+eAT7z5Xo9sZgvQ==","size":{"width":18,"height":60}} = 32

 

{"font":{"family":"Calibri","size":12,"color":"#0e101a"},"type":"$$","aid":null,"backgroundColor":"#ffffff","code":"$$\\begin{bmatrix}\n{3}&{0}&{4}\\\\\n\\end{bmatrix}$$","id":"14","backgroundColorModified":false,"ts":1647067650358,"cs":"7rK9+jQS9ehSeRLGu5eySA==","size":{"width":61,"height":16}}* {"font":{"size":12,"color":"#0e101a","family":"Calibri"},"code":"$$\\begin{bmatrix}\n{0}\\\\\n{8}\\\\\n{2}\\\\\n\\end{bmatrix}$$","aid":null,"backgroundColorModified":false,"type":"$$","backgroundColor":"#ffffff","id":"11","ts":1647067531571,"cs":"mAClrlaJyi4MMl2OSdweEw==","size":{"width":18,"height":60}} = 8

 

{"font":{"family":"Calibri","size":12,"color":"#0e101a"},"type":"$$","aid":null,"backgroundColor":"#ffffff","code":"$$\\begin{bmatrix}\n{3}&{0}&{4}\\\\\n\\end{bmatrix}$$","id":"14","backgroundColorModified":false,"ts":1647067650358,"cs":"7rK9+jQS9ehSeRLGu5eySA==","size":{"width":61,"height":16}}* {"font":{"size":12,"family":"Calibri","color":"#0e101a"},"id":"12","type":"$$","backgroundColor":"#ffffff","aid":null,"code":"$$\\begin{bmatrix}\n{1}\\\\\n{9}\\\\\n{3}\\\\\n\\end{bmatrix}$$","backgroundColorModified":false,"ts":1647067580205,"cs":"BmPU0/ACuOPyyJod54Q/9g==","size":{"width":18,"height":60}} = 15

 

{"font":{"family":"Calibri","size":12,"color":"#0e101a"},"type":"$$","aid":null,"backgroundColor":"#ffffff","code":"$$\\begin{bmatrix}\n{3}&{0}&{4}\\\\\n\\end{bmatrix}$$","id":"14","backgroundColorModified":false,"ts":1647067650358,"cs":"7rK9+jQS9ehSeRLGu5eySA==","size":{"width":61,"height":16}}* {"font":{"color":"#0e101a","family":"Calibri","size":12},"type":"$$","backgroundColorModified":false,"aid":null,"code":"$$\\begin{bmatrix}\n{2}\\\\\n{7}\\\\\n{6}\\\\\n\\end{bmatrix}$$","backgroundColor":"#ffffff","id":"13","ts":1647067608402,"cs":"VCmohIyosxF/586rat0qjg==","size":{"width":18,"height":60}} = 30

 

Hence, the resultant matrix is P2*4 = {"type":"$$","backgroundColor":"#ffffff","aid":null,"code":"$$\\begin{bmatrix}\n{12}&{8}&{11}&{11}\\\\\n{32}&{8}&{15}&{30}\\\\\n\\end{bmatrix}$$","backgroundColorModified":false,"font":{"size":16,"family":"Calibri","color":"#0e101a"},"id":"8","ts":1647067841882,"cs":"zkz5XT3i+Q2yguXkChliag==","size":{"width":152,"height":52}}

Conclusion

Matrix multiplications are useful when dealing with segments of data during image formation, Fourier transformation, and other mathematical problems. It covers a wide domain of complex algebra, trigonometry, complex numbers, and much more. With this, the major use of matrix multiplication is in the field of image processing and object transformation that requires a detailed study of each segment. For this, the matrix multiplication is divided into various complex segments that require higher-level mathematics.

 
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