Question

A square matrix $A$ is said to be an idempotent matrix if $A^2=A$. If $A,B$ and $A+B$ are idempotent matrices, then

• A. $AB=BA=0$
• B. $AB=BA=I_n$
• C. $AB=BA\ne 0$
• D. $AB=BA\ne I_n$

Answer 1

The correct option is A $AB=BA=0$

${(A+B)}^2=A^2+B^2+AB+BA\Rightarrow A+B=A+B=AB\Rightarrow AB=0$
And
${(A+B)}^2={(A-B)}^2+2AB+2BA={(A-B)}^2\Rightarrow {(A+B)}^2={(B-A)}^2\Rightarrow BA=0$