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Question
If A = `[(1, 0),(0, -1)]`, show that A3 = A.
Theorem
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Solution
A = `[(1, 0),(0, -1)]`
A2 = A × A
= `[(1 xx 1 + 0 xx 0, 1 xx 0 + 0 xx (-1)),(0 xx 1 + (-1) xx 0, 0 xx 0 + (-1) xx (-1))]`
= `[(1, 0),(0, 1)]`
= I
A3 = A2 × A
= `[(1, 0),(0, 1)] xx [(1, 0),(0, -1)]`
= `[(1 xx 1 + 0 xx 0, 1 xx 0 + 0 xx (-1)),(0 xx 1 + 1 xx 0, 0 xx 0 + 1 xx (-1))]`
= `[(1, 0),(0, -1)]`
= A
∴ A3 = A
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