Question

Prove that:
$\left|\begin{array}{ccc}13& 16& 19\\ 14& 17& 20\\ 15& 18& 21\end{array}\right|=0$

Answer 1

To prove: $\left|\begin{array}{ccc}13& 16& 19\\ 14& 17& 20\\ 15& 18& 21\end{array}\right|=0$

It can be expanded as $\left|\begin{array}{ccc}13& 16& 19\\ 14& 17& 20\\ 15& 18& 21\end{array}\right|=13(17\times 21-18\times 20)-16(14\times 21-15\times 20)+19(14\times 18-15\times 17)$

$=13(357-360)-16(294-300)+19(252-255)$

$=13(-3)-16(-6)+19(-3)$

$=-39+96-57$

$=-96+96$

Therefore, $\left|\begin{array}{ccc}13& 16& 19\\ 14& 17& 20\\ 15& 18& 21\end{array}\right|=0$