Question
Let $A,B$ and C be three $2 \times 2$ matrices with real entries such that $B = (I + A)^{- 1}$ and $A + C = I$. If $BC = \begin{bmatrix} 1 & - 5 \\ - 1 & 2 \end{bmatrix}$ and $\ CB\begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix} = \begin{bmatrix} 12 \\ - 6 \end{bmatrix},\ $ then $x_{1} + x_{2}$ is
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Solution
$B = (I + A)^{- 1},\text{ }A + C = I$
$${\Rightarrow B(I + A) = (I + A)B = I }{\Rightarrow B + BA = B + AB }{\Rightarrow B + B(I - C) = B + (I - C)B }{\Rightarrow 2\text{ }B - BC = 2\text{ }B - CB }{\Rightarrow BC = CB }{\therefore CB\begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix} = \begin{bmatrix} 1 & - 5 \\ - 1 & 2 \end{bmatrix}\begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix} = \begin{bmatrix} 12 \\ - 6 \end{bmatrix} }{\Rightarrow \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix} = \begin{bmatrix} 1 & - 5 \\ - 1 & 2 \end{bmatrix}^{- 1}\begin{bmatrix} 12 \\ - 6 \end{bmatrix} = - \frac{1}{3}\begin{bmatrix} 2 & 5 \\ 1 & 1 \end{bmatrix}\begin{bmatrix} 32 \\ - 6 \end{bmatrix} }{\Rightarrow \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix} = \begin{bmatrix} 2 \\ - 2 \end{bmatrix}\ \therefore x_{1} + x_{2} = 0}$$
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