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Addition and subtraction of matrices

If A = [aij]m × n and B = [bij]m × n are two matrices of the same order, their sum A + B is defined to be the matrix of order m × 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 ⇒ B = C

Scalar multiplication

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

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 × n and B = [bij]n × p are two matrices of order m × n and n × p respectively, then their product AB is of order m × p and is defined as
 
(AB)ij = 63317.png
= [ai1 ai2 ... ain] 63311.png
= (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 = –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 = 64534.png and B = 64528.png, then AB = 64522.pngwhile neither A nor B is the null matrix.
  • Cancellation law does not necessarily hold,i.e., if AB = AC then in general B ≠ C, even if A ≠ O.
  • Matrix multiplication A ⋅ A is represented as A2. Thus An = A ⋅ A ... n times.
  • If A = diag.(a1a2a3, ..., an) and B = diag.(b1b2b3, ..., bn), then A ⋅ 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 – 1B + nC2An – 2B2 + ... + nCnBn.
     
    Matrices A and I are always commutative. Hence, (I + A)n = nC0 + nC1A + nC1A2 +... + nCnAn.




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