Question

If $A,B$ are two idempotent matrices and
$AB=BA=0$, then $A+B$ is

• A. Scalar matrix
• B. Idempotent matrix
• C. Diagonal matrix
• D. Nilptent matrix

Answer 1

The correct option is B Idempotent matrix

Given A,B are Idempotent matrices
$A^2=A$, $B^2=B$
$(A+B{)}^2=A^2+B^2+AB+BA$
$(A+B{)}^2=A+B+AB+BA$
$(A+B{)}^2=A+B$ (since $AB=BA=0$)
Hence A+B is Idempotent