Question

If A is a square matrix such that $A^2=A$ find the value of $7A-(I+A{)}^3$.

Answer 1

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

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

$=7A-I-A^{2}A-3A-3A^2$ since $I.A=A.I=A$

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

$=7A-I-A^2-6A$

$=A-I-A$ since $A^2=A$(given)

$=-I$