Question

Calculate the values of the determinants:
$\left|\begin{array}{cccc}3& 2& 1& 4\\ 15& 29& 2& 14\\ 16& 19& 3& 17\\ 33& 39& 8& 38\end{array}\right|$.

Answer 1

$\Delta =\left|\begin{array}{cccc}3& 2& 1& 4\\ 15& 29& 2& 14\\ 16& 19& 3& 17\\ 33& 39& 8& 38\end{array}\right|C_1=C_1-{3C}_3\Rightarrow \Delta =\left|\begin{array}{cccc}0& 2& 1& 4\\ 9& 29& 2& 14\\ 7& 19& 3& 17\\ 9& 39& 8& 38\end{array}\right|C_2=C_2-{2C}_3\Rightarrow \Delta =\left|\begin{array}{cccc}0& 0& 1& 4\\ 9& 25& 2& 14\\ 7& 13& 3& 17\\ 9& 23& 8& 38\end{array}\right|C_4=C_4-{4C}_3\Rightarrow \Delta =\left|\begin{array}{cccc}0& 0& 1& 0\\ 9& 25& 2& 6\\ 7& 13& 3& 5\\ 9& 23& 8& 6\end{array}\right|$

Expanding along $R_1$

$\Rightarrow \Delta =0-0+1\left|\begin{array}{ccc}9& 25& 6\\ 7& 13& 5\\ 9& 23& 6\end{array}\right|-0\Rightarrow \Delta =\left|\begin{array}{ccc}9& 25& 6\\ 7& 13& 5\\ 9& 23& 6\end{array}\right|R_1=R_1-R_3\Rightarrow \Delta =\left|\begin{array}{ccc}0& 2& 0\\ 7& 13& 5\\ 9& 23& 6\end{array}\right|$

Expanding along $R_1$

$\Rightarrow \Delta =0-2(42-45)+0\Rightarrow \Delta =6$