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Question
If A = `[(3, 2),(7, 5)]` and B = `[(-1, -3),(5, 2)]`, verify that (AB)–1 = B–1 A–1
Solution
A = `[(3, 2),(7, 5)]`
|A| = 15 – 14
= 1 ≠ 0.A–1 exists.
adj A = `[(5, -2),(-7, 3)]`
A–1 = `1/|"A"|` adj A
= `1/1 [(5, -2),-7, 3)] = [(5, -2),(-7, 3)]`
B = `[(-1, -3),(5, 2)]`
|B| = – 2 + 15
= 13 ≠ 0.B–1 exists.
adj B = `[(2, 3),(-5, -1)]`
B–1 = `1/|"B"|` adj B
= `1/13[(2, 3),(-5, -1)]`
R.H.S : B–1A–1 = `1/13[(2, 3),(-5, -1)][(5, -2),(-7, 3)]`
= `1/13 [(10 - 211, -4 + 9),(-25 + 7, 10 - 3)]`
= `1/13 [(-11, 5),(- 18, 7)]` ..........(1)
L.H.S : AB = `[(3, 2),(7, 5)][(-1, -3),(5, 2)]`
= `[(-3 + 10, -9 + 4),(-7 + 25, -21 + 10)]`
= `[(7, -5),(8, -11)]`
|AB| = – 77 + 90
= 13 ≠ 0
(AB)–1 exists.
adj AB = `[(-11, 5),(-18, 7)]`
(AB)–1 = `1/|"AB"|` adj AB
= `1/13 [(-11, 5),(- 18, 7)]` .........(2)
(1), (2) ⇒ (AB)–1 = B–1 A–1
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