If A = [34-43]andB=[21-12], show that (A + B)(A – B) = A2 – B2 - Mathematics and Statistics

Sum

If A = `[(3, 4),(-4, 3)] and "B" = [(2, 1),(-1, 2)]`, show that (A + B)(A – B) = A² – B²

We have to prove that (A + B)(A – B) = A² – B²,

i.e., to prove A² – AB + BA – B² = A² – B²,

i.e., to prove – AB + BA = 0,

i.e., to prove AB = BA.

AB = `[(3, 4),(-4, 3)] [(2, 1),(-1, 2)]`

= `[(6 - 4, 3 + 8),(- 8 - 3, -4 + 6)]`

= `[(2, 11),(-11, 2)]` ...(i)

BA = `[(2, 1),(-1, 2)] [(3, 4),(-4, 3)]`

= `[(6 - 4, 8 + 3),(-3 - 8, -4 + 6)]`

= `[(2, 11),(-11, 2)]` ...(ii)

From (i) and (ii), we get

AB = BA

Concept: Matrices - Properties of Matrix Multiplication

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Chapter 4 Determinants and Matrices
Exercise 4.6 | Q 16 | Page 95