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प्रश्न
If A and B are square matrices such that B = − A−1 BA, then (A + B)2 = ________ .
विकल्प
O
A2 + B2
A2 + 2AB + B2
A + B
उत्तर
\[A^2 + B^2\]
\[B = - A^{- 1} BA\]
\[ \Rightarrow AB = - A A^{- 1} BA\]
\[ \Rightarrow AB = - BA . . . \left( 1 \right)\]
\[ \left( \because A A^{- 1} = I \right)\]
Now,
\[ \left( A + B \right)^2 = \left( A + B \right)\left( A + B \right)\]
\[ \Rightarrow \left( A + B \right)^2 = A^2 + AB + BA + B^2 \]
\[ \Rightarrow \left( A + B \right)^2 = A^2 - BA + BA + B^2 \left[\text{ Using }\left( 1 \right) \right]\]
\[ \Rightarrow \left( A + B \right)^2 = A^2 + B^2\]
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