If A and B are square matrices such that B = − A−1 BA, then (A + B)2 = ________ .

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KollegeApply · Accepted Answer

A^2 + B^2 B = - A^{- 1} BA \Rightarrow AB = - A A^{- 1} BA \Rightarrow AB = - BA . . . \left( 1 ight) \left( \because A A^{- 1} = I ight) Now, \left( A + B ight)^2 = \left( A + B ight)\left( A + B ight) \Rightarrow \left( A + B ight)^2 = A^2 + AB + BA + B^2 \Rightarrow \left( A + B ight)^2 = A^2 - BA + BA + B^2 \left[	ext{ Using }\left( 1 ight) ight] \Rightarrow \left( A + B ight)^2 = A^2 + B^2

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