Let's learn more about matrix multiplication

Matrix is a array of numbers or expression evaluated to numbers.The dimensions ‘m x n’ of a matrix refer to the ‘ m ‘ number of rows and ‘n’ refers to the number of rows.

MATRIX MULTIPLICATION RULES

In AB matrix multiplication , the number of columns of A must be equal to the number of rows in B. In result , the product matrix AB , will have the same number of rows as were present in matrix A and same number of columns as were present in matrix B.

MATRIX MULTIPLICATION PROPERTIES

Below are some important matrix multiplication properties

1. It does’nt support communitative property.

communitative property
2. Matrix multiplication also holds associative property.
associative property

Matrix chain multiplication , a matrix multiplication algorithm, uses this property to minimize the number of operations.

3.Distribution property of multiplication over addition also holds here 
Distribution property of multiplication

But changing the order of matrix will make this property invalid
solving matrices

4. In Square Matrix multiplication ( e.g. 2 x 2 matrix multiplication , 3 x 3 matrix multiplication , 4 x 4 matrix multiplication etc ), an identity element of same order exists. It is called either E or I

matrix chain multiplication

where A is a square matrix and I is an identity matrix

5. In scalar multiplication matrix , following is true  
scalar multiplication matrix

where x is a scalar

MATRIX MULTIPLICATION EXAMPLE

Now let’s learn how to multiply matrices from this matrix multiplication example following the matrix
multiply matrices

Solution

First , we will find the sizes of both matrices. We found 2x2 and 2x1 sizes for A and B respectively which say AB can exist as
solution for matrix multiplication

 
Size of AB will be 2x1.

Hence by
matrix multiplication example
 
State-of-the–art

Algorithms like Strassen algorithm and BLAS 3.0 based on block matrix multiplication and parallel matrix multiplication respectively are used to improve the runtime for multiplication of matrices


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