If $A$ is square matrix such that $A^{2} = I$ , then find the simplified value of $(A - I)^{3} + (A + I)^{3} - 7 A$

1 + A

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2 A

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1 - A

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A

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Solution

The correct option is D $A$
$(A - I)^{3} + (A + I)^{3} - 7 A$
$= A^{3} - 3 A^{2} I + 3 A I^{2} - I^{3} + A^{3} + 3 A^{2} I + 3 A I^{2} + I^{3} - 7 A$
$= 2 A^{3} + 6 A^{2} I - 7 A$
$= 2 A I + 6 A^{2} - 7 A$
$= 6 A^{2} - 5 A$
$= 6 I - 5 A$

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If A is a square matrix such that $A^{2} = A$ , then write the value of $7 A - (1 + A)^{3}$ , where I is an identify matrix.

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