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Adjoint of a Matrix

If A is a squ...

Question

If A is a square matrix $A^{2} = A$ show that $(I + A)^{3} = 7 A + I$

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Solution

$(I + A)^{3} - 7 A = I^{3} + A^{3} + 3 I^{2} A + 3 A^{2} I - 7 A$
$= I + A^{3} + 3 A + 3 A^{2} - 7 A$
$= I + A^{2} . A + 3 A + 3 A - 7 A$ [ $A^{2} = A$ ]
$= I + A . A - A$
$= I + A - A$
$= I$
$∴ (I + A)^{3} - 7 A = I$

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Q. If A is a square matrix such that A² = A, then (I + A)³ − 7A is equal to
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