Multiplication of Matrix

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.

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.