If A and B are square matrices such that \(B=-A^{-1} BA\), then \((A + B)^2\) is
We are given that \( B = -A^{-1}BA \), and we need to find \( (A + B)^2 \).
Start by expanding \( (A + B)^2 \):
\[
(A + B)^2 = A^2 + AB + BA + B^2.
\]
Using the given condition \( B = -A^{-1}BA \), multiply both sides by \( A \) on the left:
\[
AB = -BA.
\]
Substitute \( AB = -BA \) into the expansion of \( (A + B)^2 \):
\[
(A + B)^2 = A^2 + AB + BA + B^2 = A^2 + (-BA) + BA + B^2.
\]
The terms \( -BA + BA \) cancel out, so:
\[
(A + B)^2 = A^2 + B^2.
\]
Thus, the final result is:
\[
(A + B)^2 = A^2 + B^2.
\]
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