Question

If $A$ and $B$ are square matrices of the same order such that $A^2=A,B^2=B,AB=BA=0$, then:

• A. ${(A-B)}^2=B-A$
• B. ${(A-B)}^2=A-B$
• C. ${(A+B)}^2=A+B$
• D. None of these

Answer 1

Answer: (C)

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

Given $A^2=A$ $,$ $B^2=B$ $,$ $AB=BA=0$

$A^2+B^2=A+B$

${\left(A+B\right)}^2-2AB=A+B$ -(1)

We know $AB=0$

so put value of $AB$ in equation $\left(1\right)$

${\left(A+B\right)}^2=A+B$

Hence option (c) is correct$.$