Question

If $l$ is an identity matrix and $A$ is a square matrix such that $A^2=A$, then find the value of ${(l+A)}^2-3A$

Answer 1

$\Rightarrow$ Here, $I$ is an identity matrix, then, $I^2=I$ and $IA=A$

$\Rightarrow$ $A$ is a square matrix and $A^2=A$ [ Given ]

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

$\Rightarrow$ $I^2+2IA+A^2-3A$ [ Since, $(a+b{)}^2=a^2+2ab+b^2$ ]

$\Rightarrow$ $I+2A+A^2-3A$ [ Since, $I^2=I$ and $IA=A$ ]

$\Rightarrow$ $I+2A+A-3A$ [ Since, $A^2=A$ ]

$\Rightarrow$ $I+3A-3A$

$\Rightarrow$ $I$