Given the following matrices A 5 3 1 2 B 5 6 7 8 C 2 1 3

Given the following matrices: A = [-5 -3 -1 2] B = [-5 6 -7 8] C = [2 -1 -3 4 0 -2] D = [0 1 -3 2 -5 4] Find: AB B - A AC CD -5B DC

Solution

SOLUTION

Back-up Theory

If matrix A = ((a_ij)) is of order (m x n) and matrix B = ((b_ij)) is of order (p x q), then

1. Both (A + B) and (A - B) are defined ONLY IF m = p and n = q

2. Product AB is defined ONLY IF n = p and if this condition is satisfied, the product matrix, say C = ((c_ij)) is given by: ((c_ij)) = sum (over k = 1 to n) of (a_ik)(b_kj) or in physical terms, (i, j)th element of C is the sum of cross products of A\'s ith row elememts with B\'s jth column elements.

3.A scalar multiple of A, say kA is obtianed by multiplying each and every elemet of A by k

In the given problems, all the above conditions are fulfilled and hence all can can obtained.

Now, to get the answers, applying the above principles,

E = AB: e₁₁ = 46, e₁₂ = - 54, e₂₁ = - 9 and e₂₂ = 10

F = B - A: f₁₁ = 0, f₁₂ = 9, f₂₁ = - 6 and f₂₂ = 6

G = AC: g₁₁ = - 14, g₁₂ = 5, g₁₃ = 21, g₂₁ = 4, g₂₂ = 1 and g₂₃ = - 1

H = CD: h₁₁ = 18, h₁₂ = - 12, h₂₁ = 10 and h₂₂ = - 4

J = - 5B: j₁₁ = 25, j₁₂ = - 30, j₂₁ = 35 and j₂₂ = - 40

K = DC: k₁₁ = 4, k₁₂ = 0, k₁₃ = - 2, k₂₁ = 2, k₂₂ = 3, k₂₃ = 5, k₃₁ = 6, k₃₂ = 5, k₃₃ = 7