Find the inverse of a matrix with an unknown variable. I am trying to Express the inverse of the following matrix (assuming it exists) as a matrix containing expressions in terms of k.
$\left[ {\begin{array}{cc}
   -4 & -4 & -4    \\
   4 & 9 & 4   \\
k & -1 & 0  
  \end{array} } \right]$
Now I know that it is completely possible to solve the equation by using the conventional method for finding the inverse of a matrix. However, since there is an unknown variable, this became overly complicated to solve. Is there any other method that simplifies everything?
 A: The adjugate matrix is a way to explicitly calculate an inverse matrix from the matrix. It is most definitely not the most efficient way to do it but it is explicit. The adjugate matrix is the transpose of the cofactor matrix. In your case, we get for the adjugate matrix
$$\begin{pmatrix}
\left|\begin{matrix} 9 & 4 \\ -1 & 0\end{matrix}\right| & -\left|\begin{matrix} -4 & -4 \\ -1 & 0\end{matrix}\right| & \left|\begin{matrix} -4 & -4 \\ 9 & 4 \end{matrix}\right| \\ 
-\left|
\begin{matrix}4 & 4 \\ k & 0\end{matrix}\right| & \left|\begin{matrix} -4 & -4 \\ k & 0\end{matrix}\right| & -\left|\begin{matrix} -4 & -4 \\ 4 & 4\end{matrix}\right| \\ 
\left|\begin{matrix} 4 & 9 \\ k & -1\end{matrix}\right| & -\left|\begin{matrix} -4 & -4 \\ k & -1\end{matrix}\right| & \left|\begin{matrix} -4 & -4 \\ 4 & 9\end{matrix}\right|
\end{pmatrix}$$
Computing these we get
$$\begin{pmatrix} 4 & 4 & 20 \\ 4k & 4k & 0 \\ -4-9k & -4k-4 & -20\end{pmatrix}.$$
Finally, to get the inverse we need only to compute the determinant. Computing the determinant gives 
$$k\left|\begin{matrix} -4 & -4 \\ 9 & 4\end{matrix}\right| - (-1)\left|\begin{matrix} -4 & -4 \\ 4 & 4\end{matrix}\right| + 0\left|\begin{matrix} -4 & -4 \\ 4 & 9\end{matrix}\right| = 20 k.$$
Thus for an inverse we have
$$\frac{1}{20k}\begin{pmatrix} 4 & 4 & 20 \\ 4k & 4k & 0 \\ -4-9k & -4k-4 & -20\end{pmatrix}.$$
