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Question
If A = `[(1, 2, 2),(2, 1, 2),(2, 2, 1)]`, Show that A2 – 4A is a scalar matrix
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Solution
A2 = A · A = `[(1, 2, 2),(2, 1, 2),(2, 2, 1)] [(1, 2, 2),(2, 1, 2),(2, 2, 1)]`
= `[(1 + 4 + 4, 2 + 2 + 4, 2 + 4 + 2),(2 + 2 + 4, 4 + 1 + 4, 4 + 2 + 2),(2 + 4 + 2, 4 + 2 + 2, 4 + 4 + 1)]`
= `[(9, 8, 8),(8, 9, 8),(8, 8, 9)]`
∴ A2 – 4A = `[(9, 8, 8),(8, 9, 8),(8, 8, 9)] - 4[(1, 2, 2),(2, 1, 2),(2, 2, 1)]`
= `[(9, 8, 8),(8, 9, 8),(8, 8, 9)] -[(4, 8, 8),(8, 4, 8),(8, 8, 4)]`
= `[(5, 0, 0),(0, 5, 0),(0, 0, 5)]`
which is a scalar matrix.
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