<s>[INST] Determinate of symmetric $82\times 82$ matrix. I am trying to calculate the determinant of the matrix:
$
\left(\begin{matrix} -6 & -5 & & \dots & -5\\
-5 & -6 & \dots & & -5 \\
\vdots & & \ddots & & \vdots \\
-5 & \dots & -5 & -6 & -5\\
-5 & \dots & & -5 & -6
\end{matrix}\right)\in \mathbb{R}^{82\times 82}
$
How am I supposed to do this with a matrix so large? I tried looking for a pattern of Gauss Elimination but I got stuck. For clarification, the diagonal entries are $-6$ and the rest of the entries are $-5$. 
 [/INST] Let $M = \left(\begin{matrix} -6 & -5 & & \dots & -5\\
-5 & -6 & \dots & & -5 \\
\vdots & & \ddots & & \vdots \\
-5 & \dots & -5 & -6 & -5\\
-5 & \dots & & -5 & -6
\end{matrix}\right)$.
Then $M=\left(\begin{matrix} -5 & -5 & & \dots & -5\\
-5 & -5 & \dots & & -5 \\
\vdots & & \ddots & & \vdots \\
-5 & \dots & -5 & -5 & -5\\
-5 & \dots & & -5 & -5
\end{matrix}\right)+\left(\begin{matrix} -1 & 0 & & \dots & 0\\
0 & -1 & \dots & & 0\\
\vdots & & \ddots & & \vdots \\
0 & \dots & 0 & -1 & 0\\
0 & \dots & & 0 & -1
\end{matrix}\right)$
$=
\left(\begin{matrix} 5\\
5\\
\vdots \\
5\\
5 
\end{matrix}\right)\left(\begin{matrix} -1 & -1 & & \dots & -1\\
\end{matrix}\right)+\left(\begin{matrix} -1 & 0 & & \dots & 0\\
0 & -1 & \dots & & 0\\
\vdots & & \ddots & & \vdots \\
0 & \dots & 0 & -1 & 0\\
0 & \dots & & 0 & -1
\end{matrix}\right)$.
By the matrix determinant lemma, $\det M=\left(1+\left(\begin{matrix} -1 & -1 & & \dots & -1\\
\end{matrix}\right)\left(\begin{matrix} -1 & 0 & & \dots & 0\\
0 & -1 & \dots & & 0\\
\vdots & & \ddots & & \vdots \\
0 & \dots & 0 & -1 & 0\\
0 & \dots & & 0 & -1
\end{matrix}\right)\left(\begin{matrix} 5\\
5\\
\vdots \\
5\\
5 
\end{matrix}\right)\right)\cdot \left\vert\begin{matrix} -1 & 0 & & \dots & 0\\
0 & -1 & \dots & & 0\\
\vdots & & \ddots & & \vdots \\
0 & \dots & 0 & -1 & 0\\
0 & \dots & & 0 & -1
\end{matrix}\right\vert=(1+82\cdot5)\cdot1=411.$
</s>