I'm trying to solve a matrix equation problem and I can't work out the correct form for the equation for it to be valid.
The matrices given are:
A= $\begin{bmatrix}
1 & -1 & 3\\
4 & 1 & 5\\
0 & 0 & 0\\
\end{bmatrix}$, B= $\begin{bmatrix}
1 & -1\\
3 & 6\\
1 & 0\\
\end{bmatrix}$, C= $\begin{bmatrix}
-1 & 0\\
5 & 6\\
0 & 1\\
\end{bmatrix}$
The equation goes as follows:
$AX + B = C – X$
I arrange it to: $X= (C – B)*(A+I)^{-1}$ via the following steps:
$$AX + B = C – X$$
$$AX +X = C – B$$
$$X(A+I) = C – B /(A+I)^{-1}$$
$$X = (C – B) (A+I)^{-1}$$
But the problem is that the matrices $(C-B)$ and $(A+I)^{-1}$ can't be multiplied because they're not chained (the number of rows and collumns don't allow multiplication). I've been looking at this for over half an hour and can't figure out a different approach. Any help would be highly appreciated.
Best Answer
$X=(A+I)^{-1}(C-B )=\begin{bmatrix}\frac14 &\frac18 &\frac{-11}{8}\\\frac{-1}{2} &\frac14 &\frac14\\0 &0 &1\end{bmatrix}\begin{bmatrix}-2 &1\\2 &0\\-1 &1\end{bmatrix}=\begin{bmatrix}\frac98 &\frac{-9}{8}\\\frac54 &\frac{-1}{4}\\-1 &1\end{bmatrix}$
We get two independent solutions for $X$.