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Multiplication of Matrices

If A is a s...

Question

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

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Solution

Given $A^{2} = A$
$7 A - {(I + A)}^{3} = 7 A - (I + A) (I + 2 A I + A^{2})$
$= 7 A - (I + A) (I + 3 A) = 7 A - (I + 4 A + 3 A^{2})$
$= 7 A - I - 7 A = - I$

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