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Question
If A is a square matrix such that A2 = I, then (A – I)3 + (A + I)3 –7A is equal to ______.
Options
A
I – A
I + A
3A
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Solution
If A is a square matrix such that A2 = I, then (A – I)3 + (A + I)3 –7A is equal to A.
Explanation:
(A – I)3 + (A + I)3 – 7A = A3 – I3 – 3A2I + 3AI2 + A3 + I3 + 3A2I + 3AI2 – 7A
= 2A3 + 6AI2 – 7A
= 2A.A2 + 6AI – 7A
= 2AI + 6AI – 7A .....[A2 = I]
= 8AI – 7A
= 8A – 7A
= A
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