Maharashtra State BoardHSC Science (Electronics) 11th
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If A = [12-1-2],B=[2a-1b] and if (A + B)2 = A2 – B2 . find values of a and b - Mathematics and Statistics

Sum

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.

Concept: Matrices - Properties of Matrix Multiplication
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Balbharati Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board
Chapter 4 Determinants and Matrices
Exercise 4.6 | Q 17 | Page 95
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