The question:

Long ago, a mathematician wrote $\displaystyle C$ and $\displaystyle C^{-1}$ on a piece of paper. Unfortunately insects have damaged the paper and all that is left is:

$\displaystyle $C =

\begin{array}{ccc}

-2 & -1 & 1 \\

1 & 2 & -1 \\

* & * & * \\

\end{array}$$

$\displaystyle $C^{-1} =

\begin{array}{ccc}

* & 0 & -1 \\

2 & * & -1 \\

5 & 1 & * \\

\end{array}$$

a) Find $\displaystyle C^{-1}$

My attempt:

I tried finding the inverse of C the usual way, by substituting the asterixes with variables, and then setting up the following:

$\displaystyle $C =

\begin{array}{ccccccc}

-2 & -1 & 1 & | & 1 & 0 & 0\\

1 & 2 & -1 & | & 0 & 1 & 0\\

a & b & c & | & 0 & 0 & 1\\

\end{array}$$

I tried reducing this to row echelon form, but it quickly became a mess, and I was unable to completely reduce it. Is my approach correct so far? Thanks.