Question

Let A and B be two matrices such that AB$=$A and BA$=$B. Which of the following statements are correct?
$1$. $A^2=A$
$2$. $B^2=B$
$3$. $(AB{)}^2=AB$
Select the correct answer using the code given below.

• A. $1$ and $2$ only
• B. $2$ and $3$ only
• C. $1$ and $3$ only
• D. $1,2$ and $3$

Answer 1

The correct option is A $1$ and $2$ only

Given $AB=ABA=B$

${\left(AB\right)}^2=A^2$

$A^2={\left(AB\right)}^2$

$=A\left(BA\right)B$

$=A\left(B\right)\left(B\right)$

$=\left(AB\right)B$$=\left(AB\right)=A$

$A^2=A$ correct

$B^2={\left(BA\right)}^2$

$=B\left(AB\right)A$

$=BAA$

$=\left(BA\right)A$

$=BA$

$=B$

$\Rightarrow B^2=B$correct

${\left(AB\right)}^2=A^2=A$

${\left(AB\right)}^2\ne AB$

$\therefore$ Option A is true