Question

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

Answer 1

$7A-{(I+A)}^3$

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

$=7A-(I+A^3+3A+3A^2)$

$=7A-(I+A^2.A+3A+3A^2)$

$=7A-(I+A.A+3A+3A)$

$=7A-(I+A.A+6A)$

$=7A-(I+A^2+6A)$

$=7A-(I+A+6A)$

$=7A-(I+7A)$

$=-I$

$\therefore 7A-{(I+A)}^3=-I$