<s>[INST] Determinant of symmetric matrix Given the following matrix, is there a way to compute the determinant other than using laplace till there're $3\times3$ determinants?  
\begin{pmatrix}
  2 & 1 &1 &1&1 \\
  1 & 2 & 1& 1 &1\\
  1& 1 & 2 & 1  &1\\
 1&1 &1 &2&1\\
 1&1&1&1&-2
 \end{pmatrix}
 [/INST] Look at the matrix $$A = \begin{pmatrix} 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & -3 \end{pmatrix}.$$ It has rank two and its nonzero eigenvalues have sum $1$ (the trace) and product $$\det \begin{pmatrix} 1 & 1 \\ 1 & -3 \end{pmatrix} + \det \begin{pmatrix} 1 & 1 \\ 1 & -3 \end{pmatrix} + \det \begin{pmatrix} 1 & 1 \\ 1 & -3 \end{pmatrix} + \det \begin{pmatrix} 1 & 1 \\ 1 & -3 \end{pmatrix} = -16$$
So the characteristic polynomial is $t^3 (t^2 - t - 16)$. Evaluate this at $t = -1$ and you get $$\det(-I - A) = (-1)^5 \det \begin{pmatrix} 2 & 1 & 1 & 1 & 1 \\ 1 & 2 & 1 & 1 & 1 \\ 1 & 1 & 2 & 1 & 1 \\ 1 & 1 & 1 & 2 & 1 \\ 1 & 1 & 1 & 1 & -2 \end{pmatrix}.$$
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