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  4. If A is a square matrix such that A2=I then (A-I)3+(A+I)3-7 A s equal to .

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Q. If $A$ is a square matrix such that $A^{2}=I$ then $(A-I)^{3}+(A+I)^{3}-7 A$ s equal to _______.

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Solution:

$(A-I)^{3}+(A+I)^{3}-7 A$
$=A^{3}-I-3 A^{2}+3 A+A^{3}+I+3 A^{2}+3 A-7 A$
$=2 A^{3}-A$
$=A\left(2 A^{2}-I\right)$
$=A(2 I-I)=A$