Question

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

Answer 1

(A + B)² = A² + B²

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

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

By the equality of matrices, we get

0 = a − 2    ...(1)

0 = 1 + b    ...(2)

a + 2b = 2a − 4    ...(3)

− a − 2b = 2 + 2b    ...(4)

From equations (1) and (2), we get

a = 2 and b = − 1

The values of a and b satisfy equations (3) and (4) also.

Hence, a = 2 and b = − 1.