Introduction to multiply matrices:          
The scientist Sylvester defined  matrix is a rectangular array or arrangement of entries or elements displayed in rows and columns put within a square bracket or parenthesis. The entries or elements may be any kind of numbers like real or complex, polynomials or other expressions. Matrices are denoted by the capital letters like A, B, C.... The matrices use algebraic operations like addition, subtraction, multiplication and division. Let us see how to multiply the matrices in this article.

 

 

How to multiply matrices:

 

Let A be a matrix of order m × n and B be a matrix of order n × p then the product matrix AB will be of order m × p

i.e. order of A is m × n,order of B is n × p

Then the order of AB is m × p = number of rows of matrix A × number of columns of matrix B.

If the above mentioned condition satisfied then we can multiply matrices.

 

Examples on how to multiply matrices:

 

Ex:1 These examples give step by step procedure that shows how to multiply matrices.

Multiply 3x3 matrix A =`[[1,1,1],[2,2,2],[1,2,1]]`   with 3x2 matrix B = `[[1,2],[1,1],[2,1]]`  

Sol:

A x B = `[[1,1,1],[2,2,2],[1,2,1]]`  x   `[[1,2],[1,1],[2,1]]`  

Multiply each row of the first matrices with column of the second matrices.

=  `[[1,1,1]]``[[1],[1],[2]]``[[1,1,1]]``[[2],[1],[1]]`

`[[2,2,2]]``[[1],[1],[2]]``[[2,2,2]]``[[2],[1],[1]]`

`[[1,2,1]]``[[1],[1],[2]]``[[1,2,1]]``[[2],[1],[1]]`

multiply the first element of the left matrix with first element of the right matrix. Similarly for the second position and third position. and add those values we get,<br>

`[[1+1+4,2+1+1],[2+2+4,4+2+2],[1+2+2,2+2+1]]`          = `[[6,4],[8,8],[5,5]]`         

The resultant matrix is in 3x2 matrix.

Ex:2

Multiply 3x3 matrix A= `[[1,2,3],[4,2,1],[2,3,1]]`   with 3x3 matrix B = `[[2,6,1],[5,2,4],[1,3,3]]`

Sol:

A x B   = `[[1,2,3],[4,2,1],[2,3,1]]`  x  `[[2,6,1],[5,2,4],[1,3,3]]`

                        =  `[[1,2,3]]` `[[2],[5],[1]]` `[[1,2,3]]` `[[6],[2],[3]]` `[[1,2,3]]` `[[1],[4],[3]]`

                             `[[4,2,1]]` `[[2],[5],[1]]` `[[4,2,1]]` `[[6],[2],[3]]` `[[4,2,1]]` `[[1],[4],[3]]`

                              `[[2,3,1]]` `[[2],[5],[1]]` `[[2,3,1]]` `[[6],[2],[3]]` `[[2,3,1]]` `[[1],[4],[3]]`

                       = `[[2+10+3,6+4+9,1+8+9],[8+10+1,20+4+3,4+8+3],[4+15+1,12+6+3,2+12+3]]`

                      = `[[15,19,18],[19,27,15],[20,21,17]]`

The resultant matrix is 3x3 matrix.

Ex:3

Multiply 3x3 matrix A = `[[1,1,1],[2,2,2],[3,3,3]]` with 2x3 matrix B =  `[[1,2,3],[3,2,1]]`

Sol:

We can’t multiply these two matrixes because the number of column of the first matrix is not equal to the number of row of the second matrix.

These are the examples that shows how to multiply matrices.

 

 

Practice Problems on how to multiply matrices

 

After learning how to multiply matrix, Solve the below mentioned problems:

Ex:1

Let A =`[[2,1,4],[7,3,6]]`  and B = `[[6,4,3],[3,2,5],[7,3,1]]` are two matrices. Multiply these two matrices.

Answer: AB =  `[[43,22,15],[93,52,42]]`

Ex:2

Let A= `[[1,8],[4,3]]` and B = `[[1,3],[7,4]]`

Ans:

AB =`[[57,35],[25,24]]`