Question

If A is a square matrix such that $A^2=A$,then write the value of $7A-{\left(I+A\right)}^3$,Where i is an identify matrix

Answer 1

Given that A is an square matrix such that

$A^2=A$

Now, value of

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

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

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

$7A-(I+AA+3A+3A)$ $\because A^2=AandIA=A$

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

$7A-(I+7A)$

$7A-I-7A=-I$