Question

The value of ${\left|\begin{array}{ccc}2& 3& 7\\ 13& 17& 5\\ 15& 20& 12\end{array}\right|}^2$ is

Answer 1

Since $|AB|=|A||B|$
$|A|=\left|\begin{array}{ccc}2& 3& 7\\ 13& 17& 5\\ 15& 20& 12\end{array}\right|$
$|A|=2\left|\begin{array}{cc}17& 5\\ 20& 12\end{array}\right|-3\left|\begin{array}{cc}13& 5\\ 15& 12\end{array}\right|+7\left|\begin{array}{cc}13& 17\\ 15& 20\end{array}\right|$

$=2(17\times 12-5\times 20)-3(13\times 12-5\times 15)+7(13\times 20-15\times 17)$

$=2(204-100)-3(156-75)+7(260-255)$

$=208-243+35=0$

Now $|A{|}^2=|A|\times |A||A{|}^2=0$