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MULTIPLICATION OF MATRICES

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Multiplication of a Matrix by a Scalar

multiplicationofmatrices1

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 multiplicationofmatrices2
                              
then, 5A is obtained by multiplying each element by 5
   multiplicationofmatrices3
       

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 multiplicationofmatrices4
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.
multiplicationofmatrices5
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
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iv) Matrix multiplication is not commutative. If A and B are any two matrices then AB≠BA
                                                       

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