I have been struggling to figure out this matrix transformation question that asks me to describe a vector x that satisfies:
$$ T(x) = \begin{bmatrix} -8 \\ 9 \\ 2 \\ \end{bmatrix} $$
I am given a matrix:
$$ A = \begin{bmatrix} 1 & 3 & 1 \\ -2 & 1 & 5\\ 0 & 2 & 2\\ \end{bmatrix} $$
I am also aware that T(x) = Ax. I would like to know the general process for finding what x is when given the output vector and a matrix to be multiplied by the unknown input vector x.
$$ \begin{bmatrix} 1 & 0 & -2 & 0 \\ 0 & 1 & 1 & 0\\ 0 & 0 & 0 & 1\\ \end{bmatrix} $$
I have tried putting the augmented matrix in reduced row echelon form above, but I am not sure where to go from there. Any tips are very much appreciated.