Calculate matrix $X$ in expression $X + B = (A-B)X$

Question

I have to calculate matrix $X$ in expression $X + B = (A-B)X$.

$$ A=\left[ \begin{array} k1 & -2 & 3\\ 2 & 4 &0\\ -1 & 2 & 1\\ \end{array} \right] $$

$$ B=\left[ \begin{array} k0 & 1 & 3\\ 2 & -3 &1\\ 1 & 3 & -1\\ \end{array} \right] $$

I have no idea how to even begin here, I know basic operations with matrices and inverses, but I'm pretty much clueless here.. How do I solve this?

Answer 1

Use linearity of matrix multiplication: $$X + B = (A-B)X \\ B = (A-B)X - X = (A-B-I)X \\ X = (A-B-I)^{-1}B$$ Where $I$ is the $3\times 3$ identity matrix.

If you want to check your result, doing this calculation results in $$X = \frac16 \pmatrix{-9&-5&-15\\0&-2&-6\\-12&6&-42}$$

Answer 2

Even if $A-B-I$ were not invertible (as in the original question), we always can solve the $9$ linear equations given by the matrix equation in the $9$ variables of $X$. In the current version we obtain \begin{align*} 3x_2 & = 0, \\ 3x_5 + 1 & = 0\\ \cdots & = \cdots \\ 2x_7 + x_8 - x_9 - 1 & = 0. \end{align*}

Solving this linear system of equations gives the solution, namely $x_1=-9/6$, $x_2=0$, ..., $x_9=-42/6$. Of course, in the current version, we just have $$ X=(A-B-I)^{-1}B. $$