First the exercise asked to find the inverse matrix of the following matrix:
$$A=\begin{pmatrix}1 & 0 & 3\\
0 & 2 & 1\\
-1 & 0 & -2
\end{pmatrix}$$
I found out that:
$$
A^{-1}=0.5\begin{pmatrix}-4 & 0 & -6\\
-1 & 1 & -1\\
2 & 0 & 2
\end{pmatrix}
$$
Later they asked to solve the following equations while using that last exercise:
$$
\begin{cases}
x+3z=1\\
2y+z=2\\
-x-2z=-3
\end{cases}
$$
I don't understand how should I use the inverse matrix to solve this equation. I see that $A$ is exactly the Homogeneous system of equations. But how does it help us?
As I understand they want me to solve this system of equations without exactly solving the matrix again.
Best Answer
If $A\begin{pmatrix}x \\ y \\ z\end{pmatrix}=\begin{pmatrix}1 \\ 2 \\ -3\end{pmatrix}$, then $\begin{pmatrix}x \\ y \\ z\end{pmatrix} = A^{-1}\begin{pmatrix}1 \\ 2 \\ -3\end{pmatrix}.$