<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] You can substract the first row from every other rows and get matrix of form:
$$\begin{pmatrix}
  2 & 1 &1 &1&1 \\
  -1 & 1 & 0& 0 &0\\
  -1& 0 & 1 & 0  &0\\
 -1&0 &0 &1&0\\
 -1&0&0&0&-3
 \end{pmatrix}.$$
Computing the determinant is now much easier.
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