<s>[INST] $4$ by $4$ Determinant with variables I was given this $4$ by $4$ determinant
$$
\begin{vmatrix}
x & a & a & a \\
a & x & a & a \\
a & a & x & a \\
a & a & a & x 
\end{vmatrix} 
= 0 $$
Clearly one of the answers is $x=a,$ how do i find the other answer? 
I've tried splitting it into $4$  "$3$ by $3$" determinants but that didnt work well... 
 [/INST] Subtracting the 2nd row from the first, the 3rd from the second, the 4th from the third, tou obtain the determinant
$$\begin{vmatrix}
x-a&a-x&0&0\\0&x-a&a-x&0\\0&0&x-a&a-x\\ a&a&a&x
\end{vmatrix}=(x-a)^3\begin{vmatrix}
 1 & -1 & 0 & 0\\ 0 & 1 & -1 & 0\\ 0 & 0 & 1 & -1\\ a&a&a&x
\end{vmatrix}$$
Then, adding successively the 1st column to the first, the 2nd to the third, the 3rd to the fourth, we get an upper triangular determinant:
$$\begin{vmatrix}
 1 & -1 & 0 & 0\\ 0 & 1 & -1 & 0\\ 0 & 0 & 1 & -1\\ a&a&a&x
\end{vmatrix}=\begin{vmatrix}
 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ a&2a&3a&x+3a
\end{vmatrix}=x+3a.$$
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