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