Solving variables in a matrix for a specific determinant The matrix is as follows:
$$
A =
\begin{pmatrix}
0 & x & 1 & 2 \\
x & 1 & 1 & x \\
1 & x & x & 1 \\
1 & x & 1 & x
\end{pmatrix}
$$
What I want to do is to find all the solutions for the equation: $$\det(A) = 0$$
At first I attempted to simplify it into a polynomial, but ending up with a 4th degree term makes me wonder if there's any easier way of solving this? You can easily see that the rows/columns would be linearly independent if $x$ is equal to $1$. But I'm having a hard time realizing any other solutions this way.
Have you guys got any idea? Any help would be much appreciated!
 A: By the row operations $l_2\leftarrow l_2-xl_3$ and $l_4\leftarrow l_4-l_3$ and developing along the first column we get
$$\Delta=\det\begin{bmatrix}x&1&2\\1-x^2 &1-x^2&0\\0&1-x&x-1\end{bmatrix}$$
Now $c_2\leftarrow c_2+c_3$ and we develop along the third column we get
$$\Delta=(x-1)(x(1-x^2)-3(1-x^2)=-(x-1)^2(x-3)(x+1)$$
A: $$A =
\begin{vmatrix}
0 & x & 1 & 2 \\
x & 1 & 1 & x \\
1 & x & x & 1 \\
1 & x & 1 & x
\end{vmatrix}\stackrel{R_2-xR_3\,,R_4-R_3}=\begin{vmatrix}
0 & x & 1 & 2 \\
0 & 1-x^2 & 1-x^2 & 0 \\
1 & x & x & 1 \\
0 & 0 & 1-x & x-1
\end{vmatrix}=
$$
$$\begin{vmatrix}
 x & 1 & 2 \\
 1-x^2 & 1-x^2 & 0 \\
 0 & 1-x & x-1
\end{vmatrix}=(1-x^2)(1-x)\begin{vmatrix}
 x & 1 & 2 \\
 1 & 1 & 0 \\
 0 & 1 & \!\!-1
\end{vmatrix}=(1-x)^2(1+x)\left[-x+2+1\right]=$$
$$=-(x-1)^2(x+1)(x-3)$$
A: The most straightforward idea would be to write down your polynomial, and divide it by $x-1$ to obtain a third-degree polynomial with roots wchich are easy to guess.
Another idea would be to perform some vector/column operation on the matrix to simplify the expression for determinant from the very beginning:


*

*subtract the third line from the fourth one

*subtract the first column from the fourth one

*subtract the second column from the third one

*subtract the fourth line from the first one
Now you should obtain a rather sparce matrix, and its determinant should be easily found a factorised into a product of monimials.
If you still need help, ask in comments.
