Consider the matrix $\displaystyle \mathbf{C} = \begin{pmatrix}1 & -1\\ 1&1 \\ 0&1 \end{pmatrix}$

Find a left-inverse ofC.

Attempt:

$\displaystyle \begin{pmatrix}x1 & y1 & z1\\ x2&y2 &z2 \end{pmatrix}\begin{pmatrix}1 & -1\\ 1&1\\ 0&1 \end{pmatrix} = \begin{pmatrix}1 &0\\ 0&1\end{pmatrix}$

$\displaystyle \begin{pmatrix}x1 + y1 & -x1 +y1 + z1 \\ x2 + y2& -x2 + y2 + z2 \end{pmatrix}= \begin{pmatrix}1 &0\\ 0&1\end{pmatrix}$

I know that you suppose to somehow use gauss reduction, and thought about gauss reducing:

$\displaystyle x1 + y1 = 1$

$\displaystyle -x1 + y1 + z1 = 0$

and

$\displaystyle x2 + y2 = 0$

$\displaystyle -x2 + y2 + z2 = 1$

but didnt get any luck