<s>[INST] Invertible Matrix Let $A$ be the matrix 
$$ A=\left(\begin{array}{cccc}1&0&1&2\\2&3&\beta&4\\4&0&-\beta&-8\\ \beta&0&\beta&\beta \end{array}\right). $$
For what values of $\beta$ is the matrix invertible? 
 [/INST] $$\begin{vmatrix}
1&0&1&2\\
2&3&\beta&4\\
4&0&-\beta&-8\\
\beta&0&\beta&\beta
\end{vmatrix}
=
\beta\begin{vmatrix}
1&0&1&2\\
2&3&\beta&4\\
4&0&-\beta&-8\\
1&0&1&1
\end{vmatrix}
=
\beta\begin{vmatrix}
0&0&0&1\\
2&3&\beta&4\\
4&0&-\beta&-8\\
1&0&1&1
\end{vmatrix}
=
-\beta\begin{vmatrix}
2&3&\beta\\
4&0&-\beta\\
1&0&1
\end{vmatrix}
=3\beta
\begin{vmatrix}
4&-\beta\\
1&1
\end{vmatrix}
=3\beta(4+\beta)$$
So the determinant is zero only for $\beta=0$ and $\beta=-4$. (i.e.: these are the only cases when the matrix is singular.)
We have used the following:


*

*Effect of Elementary Row Operations on Determinant

*Laplace's Expansion Theorem

If - for some reason - you want to avoid determinants, you can simply do the elementary row operations.
(You mentioned in a comment that you haven't learned about determinants yet. It would have been better to mention this in your post.)
$$\begin{pmatrix}
1&0&1&2\\
2&3&\beta&4\\
4&0&-\beta&-8\\
\beta&0&\beta&\beta
\end{pmatrix}
\overset{(1)}\sim
\begin{pmatrix}
1&0&1&2\\
2&3&\beta&4\\
4&0&-\beta&-8\\
1&0&1&1
\end{pmatrix}
\sim
\begin{pmatrix}
0&0&0&1\\
2&3&\beta&4\\
4&0&-\beta&-8\\
1&0&1&1
\end{pmatrix}
\sim
\begin{pmatrix}
1&0&1&1\\
2&3&\beta&4\\
4&0&-\beta&-8\\
0&0&0&1
\end{pmatrix}
\sim
\begin{pmatrix}
1&0&1&1\\
0&3&\beta-2&2\\
0&0&-\beta-4&-12\\
0&0&0&1
\end{pmatrix}
\sim
\begin{pmatrix}
1&0&1&1\\
0&3&\beta-2&0\\
0&0&-\beta-4&0\\
0&0&0&1
\end{pmatrix}
\overset{(2)}\sim
\begin{pmatrix}
1&0&1&0\\
0&3&\beta-2&0\\
0&0&1&0\\
0&0&0&1
\end{pmatrix}
\sim
\begin{pmatrix}
1&0&0&0\\
0&3&0&0\\
0&0&1&0\\
0&0&0&1
\end{pmatrix}
\sim
\begin{pmatrix}
1&0&0&0\\
0&1&0&0\\
0&0&1&0\\
0&0&0&1
\end{pmatrix}
$$
You see that the matrix is invertible. But we have to be careful about steps $(1)$ and $(2)$. In $(1)$ we have divided by $\beta$ and in $(2)$ we have divided by $-\beta-4$. So these steps are valid only for $\beta\ne 0,4$. If you try elimination for $\beta=0$ and $\beta=4$, you will find out that for these values the matrix is singular.
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