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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
Solution
\[\left( A - I \right)^3 + \left( A + I \right)^3 - 7A\]
\[ = A^3 - I^3 - 3 A^2 I + 3A I^2 + A^3 + I^3 + 3 A^2 I + 3A I^2 - 7A\]
\[ = 2 A^3 + 6A I^2 - 7A\]
\[ = 2A . A^2 + 6A - 7A\]
\[ = 2A . I - A \left( \because A^2 = I \right)\]
\[ = 2A - A\]
\[ = A\]
Hence, the correct option is (a).
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