Question

If A and B are two square matrices such that $B=-A^{-1}$ BA, then $(A+B{)}^2=$

• A. $0$
• B. $A^2+B^2$
• C. $A^2+2AB+B^2$
• D. $A+B$

Answer 1

The correct option is B $A^2+B^2$

Given,$B=-A^{-1}BA\therefore AB=-A{A}^{-1}BA=-IBA=-BA\therefore AB=-BA$
Now $(A+B{)}^2=(A+B\left)\right(A+B)=A^2+AB+BA+B^2$
$=A^2+B^2[BA=-AB]$
Thus, $(A+B{)}^2=A^2+B^2$.