# Multiplication of a Matrix by a Scalar

When a matrix is multiplied by a scalar then each element of that matrix is multiplied by the scalar. In general, we can say, if A = [aij]m x n is a matrix and ‘k’ is a scalar, then kA is another matrix which is obtained by multiplying each element of A by ‘k’.

For example: If

then, 5A is obtained by multiplying each element by 5

## Properties of Scalar Multiplication

i) If A and B are matrices and k is a scalar then k(A + B) = kA + kB

ii) If A is a matrix and k and l are scalars then (k + l) A = kA + lA

## Multiplication of Matrices

The product of two matrices A and B is defined if the number of columns of A is equal to the number of rows of B. Let A = [aij] be m x n matrix and B = [bjk] be an n x p matrix. Then, the product of A and B is a matrix C of order m x p.

For example: Find AB, if

Solution: Matrix A has 2 columns and B has 2 rows, so number of columns of 1st matrix is same as number of rows of 2nd matrix, hence it is conformable for multiplication.

Order of A = 2 x 2 and order of B = 2 x 3, so order of AB = 2 x 3

## Properties of Multiplication of Matrices

i) Associative Law: For any three matrices A, B and C, we have

(AB)C = A(BC), whenever both sides of equality are defined.

ii) Distributive law: For three matrices A, B and C

a) A (B + C) = AB + AC

b) (A + B) C = AC + BC

iii) Existence of multiplicative identity: For every square matrix A, there exists an identity matrix of same order such that IA = AI = A

iv) Matrix multiplication is not commutative. If A and B are any two matrices then AB≠BA

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### Reference Links:

- http://en.wikipedia.org/wiki/Matrix_(mathematics)
- http://en.wikipedia.org/wiki/Scalar_(mathematics)
- http://www.mathwords.com/r/row_of_a_matrix.htm
- http://www.mathwords.com/c/column_of_a_matrix.htm
- http://en.wikipedia.org/wiki/Associativity
- http://en.wikipedia.org/wiki/Distributivity
- http://dictionary.reference.com/browse/multiplicative+identity