# Addition and subtraction of matrices

If A = [aij]m Ã— n and B = [bij]m Ã— n are two matrices of the same order, their sum A + B is defined to be the matrix of order m Ã— n such that (A + B)ij = aij + bij for i = 1, 2, ..., m and j = 1, 2, ..., n

Notes:
• Only matrices of the same order can be added or subtracted.
• Addition of matrices is commutative as well as associative.
• Cancellation laws hold well in case of addition.
• That is , A + B = A + C â‡’ B = C

# Scalar multiplication

The matrix obtained by multiplying every element of a matrix A by a scalar Î» is called the scalar multiple of A by Î» and is denoted by Î» A, i.e., if A = [aij] then Î»A = [Î»aij].

# Multiplication of matrices

Two matrices A and B are conformable for the product AB if the number of columns in A (pre-multiplier) is same as the number of rows in B (post-multiplier). Thus, if A = [aij]m Ã— n and B = [bij]n Ã— p are two matrices of order m Ã— n and n Ã— p respectively, then their product AB is of order m Ã— p and is defined as

(AB)ij =
= [ai1 ai2 ... ain]
= (ith row of A) (jth column of B) ...(1)

i = 1, 2, ..., m and j = 1, 2, ..., p

Notes:
• Commutative law does not necessarily hold for matrices.
• If AB = BA then matrices A and B are called commutative matrices.
• If AB = â€“BA then matrices A and B are called anti-commutative matrices.
• Matrix multiplication is associative A(BC) = (AB)C.
• Matrix multiplication is distributive with respect to addition, i.e., A(B + C) = AB +AC.
• The matrices possess divisors of zero, i.e.,if the product AB = O, it is not necessary that at least one of the matrices should be zero matrix. For example, if A =  and B = , then AB = while neither A nor B is the null matrix.
• Cancellation law does not necessarily hold,i.e., if AB = AC then in general B â‰  C, even if A â‰  O.
• Matrix multiplication A â‹… A is represented as A2. Thus An = A â‹… A ... n times.
• If A = diag.(a1a2a3, ..., an) and B = diag.(b1b2b3, ..., bn), then A â‹… B = diag.(a1b1,a2b2, ..., anbn). Thus An = dign.(a1na2n,a3n, ..., ann)
• If A and B are diagonal matrices of the same order then AB = BA or diagonal matrices are commutative.
• If A and B are commutative then

(A + B)2 = (A + B)(A + B)

A2 + AB + BA + B2g

A2 + 2AB + B2

Similarly, (A + B)3 = A3 + 3A2B + 3AB2 + B3.

In general, (A + B)n = nC0An + nC1An â€“ 1B + nC2An â€“ 2B2 + ... + nCnBn.

Matrices A and I are always commutative. Hence, (I + A)n = nC0 + nC1A + nC1A2 +... + nCnAn.