Let A,B,C be three 22 matrices with real entries such that B=(I+A)^-1 and A+C=I.

Question

Let $A,B,C$ be three $2	imes2$ matrices with real entries such that $B=(I+A)^{-1}$ and $A+C=I$. If $BC=\begin{pmatrix}1&-5\-1&2\end{pmatrix}$ and $CB\begin{pmatrix}x_1\x_2\end{pmatrix}=\begin{pmatrix}12\-6\end{pmatrix}$, then $x_1+x_2$ is:

Accepted Answer

From $B=(I+A)^{-1}$: $B^{-1}=I+A$. From $A+C=I$: $A=I-C$, so $B^{-1}=2I-C$.

Multiplying $B\cdot B^{-1}=I$ pre- and post- by appropriate factors gives $BC=2B-I$ and $CB=2B-I$, so $BC=CB$.

Hence $CB=\begin{pmatrix}1&-5\-1&2\end{pmatrix}$, and the system $CB\begin{pmatrix}x_1\x_2\end{pmatrix}=\begin{pmatrix}12\-6\end{pmatrix}$ gives
$$x_1-5x_2=12,\qquad -x_1+2x_2=-6.$$

Adding: $-3x_2=6\Rightarrow x_2=-2$. Then $-x_1+2(-2)=-6\Rightarrow x_1=2$.

So $x_1+x_2=2+(-2)=0$.

Solutions · 1