Question

If A = `[(1, 2),(-1, -2)], "B" = [(2, "a"),(-1, "b")]` and (A + B)² = A² + B², find the values of a and b.

Answer 1

Given (A + B)² = A² + B²

∴ A² + AB + BA + B² = A² + B²

∴ AB + BA = 0

∴ AB = – BA

∴ `[(1, 2),(-1, 2)] [(2, "a"),(-1, "b")]  = - [(2, "a"),(-1, "b")][(1, 2),(-1, -2)]`

∴ `[(2 - 2, "a" + 2"b"),(-2 + 2 ,-"a" - 2"b")] = -[(2 - "a", 4-2"a"),(-1 - "b", -2 -2"b")]`

∴ `[(0, "a" + 2"b"),(0, -"a"- 2"b")] = [(-2 + "a" , -4 + 2"a"),(1 + "b", 2 + 2"b")]`

∴ By equality of matrices, we get

– 2 + a = 0 and 1 + b = 0

∴ a = 2 and b = – 1.