Question

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

पर्याय

• O

• A² + B²

• A² + 2AB + B²

• A + B

Answer 1

\[A^2 + B^2\]

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