Question

If $A,B,A+I,A+B$ are idempotent matrices, then $AB$ is equal to

• A. $BA$
• B. $-BA$
• C. $I$
• D. $0$

Answer 1

The correct option is B $-BA$

Given, $A,B,A+I,A+B$ are idempotent matrices. i.e.,

$A^2=A$,

$B^2=B$,

$(A+I{)}^2=A+I$ and

$(A+B{)}^2=A+B$

$\Rightarrow (A+B)\times (A+B)=A+B$

$\Rightarrow (A+B)\times A+(A+B)\times B=A+B$

$\Rightarrow A^2+BA+AB+B^2=A+B$

$\Rightarrow A+B+AB+BA=A+B$

Therefore, we have

$AB+BA=0$

$\Rightarrow AB=-BA$

Hence. the correct option is $\left(B\right)$.