Question

Find the value of $\left|\begin{array}{ccc}29& 26& 22\\ 25& 31& 27\\ 63& 54& 46\end{array}\right|$.

Answer 1

The given determinant

$\left|\begin{array}{ccc}29& 26& 22\\ 25& 31& 27\\ 63& 54& 46\end{array}\right|$

$C_1\to C_1-C_2$ and $C_3\to C_3-C_2$
$=\left|\begin{array}{ccc}3& 26& -4\\ -6& 31& -4\\ 9& 54& -8\end{array}\right|$

Taking common $3$ from $C_1$ and $-4$ from $C_3$

$=-3\times 4\times \left|\begin{array}{ccc}1& 26& 1\\ -2& 31& 1\\ 3& 54& 2\end{array}\right|$

$R_2\to R_2-R_1$ and $R_3\to R_3-2R_1$

$=-12\times \left|\begin{array}{ccc}1& 26& 1\\ -3& 5& 0\\ 1& 2& 0\end{array}\right|$
$=-12\left|\begin{array}{ccc}1& 1& 26\\ 0& -3& 5\\ 0& 1& 2\end{array}\right|=-12\left|\begin{array}{cc}-3& 5\\ 1& 2\end{array}\right|=132$.