If A and B are square matrices

Question:

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

(a) 0

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

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

(d) $A+B$

Solution:

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

$B=-A^{-1} B A$

$\Rightarrow A B=-A A^{-1} B A$

$\Rightarrow A B=-B A$            ....(1)

$\left(\because A A^{-1}=I\right)$

Now,

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

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

$\Rightarrow(A+B)^{2}=A^{2}-B A+B A+B^{2}$      [Using (1)]

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

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