Question

If A is a square matrix such that $A^2=A$, then write the value of $7A-(1+A{)}^3$, where I is an identify matrix.

Answer 1

We have, $7A-(I+A{)}^3=7A-(I+A\left)\right[(I+A)(I+A)]=7A-(I+A\left)\right[I.I+I.A+A.I+A.A]$

$\Rightarrow 7A-(I+A)[I+2A+A]\{\because A.A=A^2=A,A.I=A=I.A\}$

$\Rightarrow =7A-(I+A)(I+3A)=7A-(I+4A+3A^2)=-I$. [Given that $A^2=A$].