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Question
If A = `[(1, 2),(3, 4)]` verify that A (adj A) = (adj A) A = |A| I
Solution
For A = `[(1, 2),(3, 4)]`
A11 = (–1)1 + 1 (4) = 4
A12 = (–1)1 + 2 (3) = –3
A21 = (–1)2 + 1 (2) = –2
A22 = (–1)2 + 2 (1) = 1
adj A = `[(A_11, A_21),(A_12, A_22)]`
= `[(4, -2),(-3, 1)]`
∴ A(adj A) = `[(1, 2),(3, 4)][(4, -2),(-3, 1)]`
= `[(4 - 6, -2 +2),(12 -12, -6 + 4)]`
= `[(-2, 0),(0, -2)]` ...(i)
(adj A) . A = `[(4, -2),(-3, 1)][(1, 2),(3, 4)]`
= `[(4 - 6, 8 - 8),(-3 + 3, -6 + 4)]`
= `[(-2, 0),(0, -2)]` ...(ii)
and |A| I = `[(1, 2),(3, 4)][(1, 0),(0, 1)]` = 4 -6 = -2
= `(-2) [(1, 0),(0, 1)]`
= `[(-2, 0),(0, -2)]` ...(iii)
From (i), (ii) and (iii) we get,
A(adj A) = (adj A) A = |A| I
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