If $$ A $$ and $$ B $$ are square matrices such that $$ |A|
eq 0,|B|
eq 0 $$ and $$ B=-A^{-1} B A $$ , then $$ (A+B)^{2}=\ldots $$ $$ \begin{array}{llll} ext { (a) } A^{2}+B^{2} & ext { (b) } 0 & ext { (c) } A^{2}+2 A B+B^{2} & ext { (d) } A+B\end{array} $$

Question

If $$ A $$ and $$ B $$ are square matrices such that $$ |A| 
eq 0,|B| 
eq 0 $$ and $$ B=-A^{-1} B A $$ , then $$ (A+B)^{2}=\ldots $$ $$ \begin{array}{llll}	ext { (a) } A^{2}+B^{2} & 	ext { (b) } 0 & 	ext { (c) } A^{2}+2 A B+B^{2} & 	ext { (d) } A+B\end{array} $$

UpStudy AI · Accepted Answer

We are given that  
$$
B = -A^{-1}BA.
$$  
Multiplying both sides on the left by $$A$$ (which is allowed since $$|A|\neq 0$$ so $$A$$ is invertible), we obtain  
$$
AB = -BA.
$$

Now, consider expanding $$(A+B)^2$$:
$$
(A+B)^2 = A^2 + AB + BA + B^2.
$$
Since we found that $$AB = -BA$$, the middle terms cancel:
$$
AB + BA = -BA + BA = 0.
$$
Thus,
$$
(A+B)^2 = A^2 + B^2.
$$

The final answer is $$\text{(a)}\ A^2+B^2$$.
$$(A + B)^2 = A^2 + B^2$$, so the answer is (a) $$A^2 + B^2$$.

If and are square matrices such that and
, then

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