Question

Evaluate the determinants

(i)$\left|\begin{array}{ccc}3& -1& -2\\ 0& 0& -1\\ 3& -5& 0\end{array}\right|$

(ii)$\left|\begin{array}{ccc}0& 1& 2\\ -1& 0& -3\\ -2& 3& 0\end{array}\right|$

(iii)$\left|\begin{array}{ccc}3& -4& 5\\ 1& 1& -2\\ 2& 3& 1\end{array}\right|$

(iv)$\left|\begin{array}{ccc}2& -1& -2\\ 0& 2& -1\\ 3& -5& 0\end{array}\right|$

Answer 1

(i)Let $\left|\begin{array}{ccc}3& -1& -2\\ 0& 0& -1\\ 3& -5& 0\end{array}\right|$

It can be observed that in the second row, two entries are zero. Thus, we expand along the second row for easier calculation.

$|A|=-0\left|\begin{array}{cc}-1& -2\\ -5& 0\end{array}\right|+0\left|\begin{array}{cc}3& -2\\ 3& 0\end{array}\right|-(-1)\left|\begin{array}{cc}3& -1\\ 3& -5\end{array}\right|$

$=1(-15+3)$

$=-12$

(ii)Let $\left|\begin{array}{ccc}0& 1& 2\\ -1& 0& -3\\ -2& 3& 0\end{array}\right|$

By expanding along the first row, we have:

$|A|=3\left|\begin{array}{cc}1& -2\\ 3& 1\end{array}\right|+4\left|\begin{array}{cc}1& -2\\ 2& 1\end{array}\right|+5\left|\begin{array}{cc}1& 1\\ 2& 3\end{array}\right|$

$=3(1+6)+4(1+4)+5(3-2)$

$=21+20+5$

$=46$

(iii)Let $\left|\begin{array}{ccc}3& -4& 5\\ 1& 1& -2\\ 2& 3& 1\end{array}\right|$

By expanding along the first row, we have:

$|A|=0\left|\begin{array}{cc}0& -3\\ 3& 0\end{array}\right|-1\left|\begin{array}{cc}-1& -2\\ -3& 0\end{array}\right|+2\left|\begin{array}{cc}-1& 0\\ -2& 3\end{array}\right|$

$=0-1(0-6)+2(-3-0)$

$=6-6$

$=0$

(iv)Let $\left|\begin{array}{ccc}2& -1& -2\\ 0& 2& -1\\ 3& -5& 0\end{array}\right|$

By expanding along the first column, we have:

$|A|=2\left|\begin{array}{cc}2& -1\\ -5& 0\end{array}\right|-0\left|\begin{array}{cc}-1& -2\\ -5& 0\end{array}\right|+3\left|\begin{array}{cc}-1& -2\\ 2& -1\end{array}\right|$

$=2(0-5)-0+3(1+4)$

$=-10+15$

$=5$