If AB = A and BA = B, where A and B are square matrices, then


If $A B=A$ and $B A=B$, where $A$ and $B$ are square matrices, then

(a) $B^{2}=B$ and $A^{2}=A$

(b) $B^{2} \neq B$ and $A^{2}=A$

(c) $A^{2} \neq A, B^{2}=B$

(d) $A^{2} \neq A, B^{2} \neq B$


(a) $B^{2}=B$ and $A^{2}=A$


$A B=A \quad \ldots(1)$

$B A=B \quad \ldots(2)$

$\Rightarrow A B A=A A \quad$ [Multiplying both sides by $A$ ]

$B A B=B B$       [Multiplying both sides by $A$ ]

$\Rightarrow A B=A^{2} \quad[$ From eq. (2) $]$

$B A=B^{2} \quad$ [From eq. (1) $]$

$\Rightarrow A=A^{2} \quad[$ From eq. $(1)]$

$B=B^{2} \quad$ [From eq. (2)]



Leave a comment


Click here to get exam-ready with eSaral

For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.

Download Now