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Question
If \[A = \begin{bmatrix}1 & - 1 \\ 2 & - 1\end{bmatrix} \text { and } B = \begin{bmatrix}a & 1 \\ b & - 1\end{bmatrix} \text { and } \left( A + B \right)^2 = A^2 + B^2\] , then find the values of a and b.
Solution
Given: \[\left( A + B \right)^2 = A^2 + B^2\]
\[\Rightarrow \left( A + B \right)\left( A + B \right) = A^2 + B^2 \]
\[ \Rightarrow A\left( A + B \right) + B\left( A + B \right) = A^2 + B^2 \]
\[ \Rightarrow A^2 + AB + BA + B^2 = A^2 + B^2 \]
\[ \Rightarrow AB + BA = O\]
So,
\[\begin{bmatrix}1 & - 1 \\ 2 & - 1\end{bmatrix}\begin{bmatrix}a & 1 \\ b & - 1\end{bmatrix} + \begin{bmatrix}a & 1 \\ b & - 1\end{bmatrix}\begin{bmatrix}1 & - 1 \\ 2 & - 1\end{bmatrix} = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}a - b & 2 \\ 2a - b & 3\end{bmatrix} + \begin{bmatrix}a + 2 & - a - 1 \\ b - 2 & - b + 1\end{bmatrix} = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}2a - b + 2 & - a + 1 \\ 2a - 2 & - b + 4\end{bmatrix} = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\]
\[ \Rightarrow 2a - b + 2 = 0, - a + 1 = 0, 2a - 2 = 0, - b + 4 = 0\]
\[ \Rightarrow a = 1, b = 4\]
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