Question

If $A$ and $B$ are two matrices such that $AB=A$ and $BA=B$, then $B^2$ is equal to

• A. $B$
• B. $A$
• C. $1$
• D. $0$

Answer 1

The correct option is A $B$

Given for $A$ and $B$ are two matrices such that $AB=A$.....(1) and $BA=B$......(2).

Now, from (2) we have,

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

$\Rightarrow B^2=B\left(AB\right)A$ [ Since matrix multiplication is associative]

$\Rightarrow B^2=B\left(A\right)A$ [ Using (1)]

$\Rightarrow B^2=\left(BA\right)A$[ Since matrix multiplication is associative]

$\therefore B^2=BA=B$ [ Using (2)]