If A = `[(1, 2),(-1, -2)], "B" = [(2, "a"),(-1, "b")]` and if (A + B)^{2} = A^{2} – B^{2} . find values of a and b

#### Solution

Since (A+ B)^{2} =A^{2} + B^{2},

(A + B)(A + B) = A^{2} + B^{2}

∴ A^{2} + AB + BA + B^{2} = A^{2} + B^{2}

∴ 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 the equality of matrices,

0 = – 2 + a ... (1)

0 = 1 + b ... (2)

a + 2b = – 4 + 2a ... (3)

– a – 2b = 2 + 2b ... (4)

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

a = 2 and b= – 1

Since the values of a and b satisfy equations (3) and (4).

Hence, a = 2 and b = – 1.