Finding a General Solution to a Nonhomogeneous Matrix Equation

Question

I've come across a question in which I've been asked to find the general solution to the matrix equation:

$\begin{bmatrix}1 & -2 & 1\\-2 & 4 &-2\\1 & -2 & 1\end{bmatrix} \begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix}8\\-16\\8\end{bmatrix}$

by first finding a solution to a similar homogeneous equation:

$\begin{bmatrix}1 & -2 & 1\\-2 & 4 &-2\\1 & -2 & 1\end{bmatrix} \begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix}0\\0\\0\end{bmatrix}$

and then using that along with a given solution,

\begin{bmatrix}1\\-2\\3\end{bmatrix}

to find the general solution.

I've been looking around online and in textbooks for awhile and can't seem to find any information on this type of problem. Could someone provide a rundown of the methodology?

Answer 1

General solution = (particular solution) + (general solution of the homogeneous equation).