I have a surjective matrix such that
$$A * V = A * [1,1,1]$$
$$A = \begin{bmatrix}1&2&3\\2&1&1\end{bmatrix}$$
I need to find V that is of shape (3,).
This is what I did so far
$$\begin{bmatrix}1&2&3\\2&1&1\\0&0&1\end{bmatrix}.\begin{bmatrix}v_1\\v_2\\v_3\end{bmatrix}=\begin{bmatrix}6\\4\\?\end{bmatrix}$$
I used a value of 2 for the ?
So
$$\begin{bmatrix}1&2&3\\2&1&1\\0&0&1\end{bmatrix}.\begin{bmatrix}v_1\\v_2\\v_3\end{bmatrix}=\begin{bmatrix}6\\4\\2\end{bmatrix}$$
Solving this got me
$$v_1 + 2v_2 + 3v_3 = 6$$
$$2v_1 + v_2 + v_3 = 4$$
$$v_3 = 2$$
which implies
$$v_1 = -2, v_2 = 6, v_3 = 2$$
But apparently, this is wrong. I am not sure what I am doing wrong. Any help is appreciated. Thanks in advance
Best Answer