Multiplication of Matrices

A matrix is said to be as ordered rectangular array of number. The operation on matrices that is the multiplication of a matrix generally falls into two categories

1. Scalar Multiplication: In the matrix, a real number is called a scalar in which a single number is being multiplied by all the elements present in the matrix.
2. Multiplication of the matrix with another entire matrix.

Scalar Multiplication

Multiplication of scalar means, multiplying a matrix by a number i.e. a real number. In general, we may define multiplication of a matrix by a scalar as follows: If  is a matrix and k is a scalar, then kA is another matrix which is obtained by multiplying each element of A by the scalar number k. In other words,  that is, (i,j) element of kA is  for all possible values of i and j.

Example 1: Suppose Mohan has a factory which produces clothes for boys and girls in three different price tags. The quantity produced by a factory is represented as matrices given below :   . Now Mohan wants to doubled the production for all three different prices tags clothes. Represent it in a matrix form.

Solution: From the above information the matrix can be represented by doubling each price categories as follows: and 

Total Production =  

Example 2: If . Find 4A.

Solution :  = 

Multiplication between two matrices

Multiplication between two matrices is feasible if the number of columns of the first matrix is same as the matrix of rows in another matrix then matrix multiplication can be done. In general, Let  be an m*n matrix and  be an n*p matrix. Then the product of the matrices A and B is the matrix C of order m*p. To get the  element   of the matrix C, we take the  row of A and  column of B, multiply then element wise and take sum of all these products. In other words, If ,  then the  row of A is  and the  column of B is , then  = . The matrix  is the product of A and B.

Properties of multiplication of matrices

1. The associative law: For any three matrices A, B and C. We have (AB)C =A(BC), whenever both sides of the equality are defined.
2. The distributed law: For three matrices A, B and C.  (i) A (B+C) = AB + AC and (ii) (A+B)C = AC + BC, whenever both sides of equality are defined.
3. The existence of multiplicative identity: For every square matrix A, there exists an identity matrix of same order such that IA = AI = A.

Example 3: If  and . Find product of AB.

Solution: This is a 2*2 matrix in which each entry in the sum of the product across some row of the A with the corresponding entries down some columns of B. These four computations is:

Entry in first row first column  = 

Entry in first row second column  = 

Entry in second row first column  = 

Entry in second row second column  = 

Thus 

Exercise

1. If , Find 2A
2. Let  and . Find AB and BA.
3. If  and . Find AB.
4. Show that the matrix  satisfies the equation .
5. If  and . Find AB.