# Math Help - Finding the inverse of a matrix with missing values

1. ## Finding the inverse of a matrix with missing values

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

$C =
\begin{array}{ccc}
-2 & -1 & 1 \\
1 & 2 & -1 \\
* & * & * \\
\end{array}$

$C^{-1} =
\begin{array}{ccc}
* & 0 & -1 \\
2 & * & -1 \\
5 & 1 & * \\
\end{array}$

a) Find $C^{-1}$

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

$C =
\begin{array}{ccccccc}
-2 & -1 & 1 & | & 1 & 0 & 0\\
1 & 2 & -1 & | & 0 & 1 & 0\\
a & b & c & | & 0 & 0 & 1\\
\end{array}$

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.

2. Remember that the inverse of the matrix is given by 1/|C| * adj(C)

So if we find all the cofactors of the matrix C, and then equate them to the known values of C^{-1}, we can then solve using simultaneous equations.

Does this make sense?

3. Give a symbol to each unknown value

$C = \left[\begin{matrix}-2&-1&\phantom{-}1\\ \phantom{-}1&\phantom{-}2&-1\\ \phantom{-}a&\phantom{-}b&\phantom{-}c\end{matrix}\right]$

$C^{-1} = \left[\begin{matrix}\phantom{-}d&\phantom{-}0&-1\\\phantom{-}2&\phantom{-}e&-1\\ \phantom{-}5&\phantom{-}1&\phantom{-}f\end{matrix}\right]$.

You should know that when you multiply a matrix by its inverse, you get the identity matrix.

So $\left[\begin{matrix}-2&-1&\phantom{-}1\\ \phantom{-}1&\phantom{-}2&-1\\ \phantom{-}a&\phantom{-}b&\phantom{-}c\end{matrix}\right] \left[\begin{matrix}\phantom{-}d&\phantom{-}0&-1\\\phantom{-}2&\phantom{-}e&-1\\ \phantom{-}5&\phantom{-}1&\phantom{-}f\end{matrix}\right] = \left[\begin{matrix}1&0&0\\0&1&0\\0&0&1\end{matrix}\righ t]$.

Perform the multiplication and you should be able to find the unknowns.

4. Thank you soooo much! I've been stuck on this question for two days!