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The value of the determinant $\left|\begin{array}{ccc}31 & 37 & 92 \\ 31 & 58 & 71 \\ 31 & 105 & 24\end{array}\right|$ is
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$\left|\begin{array}{ccc}31 & 37 & 92 \\ 31 & 58 & 71 \\ 31 & 105 & 24\end{array}\right|=\left|\begin{array}{ccc}31 & 37 & 92 \\ 0 & 21 & -21 \\ 0 & 47 & -47\end{array}\right|$ ;$\begin{aligned} & R_3 \rightarrow R_3-R_2 \\ & R_2 \rightarrow R_2-R_1\end{aligned}$
$=\left|\begin{array}{ccc}31 & 129 & 92 \\ 0 & 0 & -21 \\ 0 & 0 & -47\end{array}\right|=0$; (by $C_2 \rightarrow C_2+C_3$ ).
$=\left|\begin{array}{ccc}31 & 129 & 92 \\ 0 & 0 & -21 \\ 0 & 0 & -47\end{array}\right|=0$; (by $C_2 \rightarrow C_2+C_3$ ).
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