Find the set of all $\alpha$ such that Matrix A is invertible and calculate the inverse for all $\alpha$ $A=\begin{pmatrix}
        0 & 1 & -1 & 2\\
        2 & -1 & 3 & 0 \\
        \alpha & 0 & 1& 0 \\ 
3 & -1 &4 & 0
        \end{pmatrix}$
I know that a Matrix is invertible only if $det(A)\not=0$
$det(A)=0\begin{vmatrix}
        -1 & 3 & 0 \\
        0 & 1 & 0 \\
        -1 & 4 & 0 \\
        \end{vmatrix}-1\begin{vmatrix}
        2 & 3 & 0 \\
        \alpha & 1 & 0 \\
        3 & 4 & 0 \\
        \end{vmatrix}-1\begin{vmatrix}
        2 & -1 & 0 \\
        \alpha & 0 & 0 \\
        3 & -1 & 0 \\
        \end{vmatrix}-2\begin{vmatrix}
        2 & -1 & 3 \\
        \alpha & 0 & 1 \\
        3 & -1 & 4 \\
        \end{vmatrix}$
$=-[2\begin{vmatrix}
        1 & 0\\
        4 & 0\\
        \end{vmatrix}-3\begin{vmatrix}
        \alpha & 0\\
        3 & 0\\
        \end{vmatrix}]-[2\begin{vmatrix}
        0 & 0\\
        -1 & 0\\
        \end{vmatrix}+\begin{vmatrix}
        \alpha & 0\\
        3 & 0\\
        \end{vmatrix}]-2[2\begin{vmatrix}
        0 & 1\\
        -1 & 4\\
        \end{vmatrix}+\begin{vmatrix}
        \alpha & 1\\
        3 & 4\\
        \end{vmatrix}+3\begin{vmatrix}
        \alpha & 0\\
        3 & -1\\
        \end{vmatrix}]$
$=-2(2+4\alpha-3-3\alpha)=-4-8\alpha+6+6\alpha=-2\alpha+2$
$-2\alpha+2=0 \iff \alpha=1$
So the matrix is invertible for all $\alpha \in \Bbb R| \alpha\not=0$
But how do I calculate the inverse of A for ALL other $\alpha$ without spending the rest of my life on this question? Am I missing something here?
Thanks in advance?

The determinant of $A$ is given by $\det(A)=2(1-a)$, and the inverse, for all $a\neq 1$, is given by
$$
A^{-1}=
\frac{1}{\det(A)}\begin{pmatrix} 0 & -2 & -2 & 2 \cr 
0 & 2(4a-3) & 2 & 2(2-3a) \cr
0 & 2a & 2 & -2a \cr
1-a & 3(1-a) & 0 & 2(a-1)
\end{pmatrix}.
$$
You don't need to spend the rest of your life for its computation. For example, the Gauss elimination with $A$ in parallel with the identity matrix, gives you the result fairly easy. Also, it is useful to get familiar with a computer algebra system "early in life". Almost all people I know do no longer compute inverse matrices of size $4$ by hand.
