Question

Evaluate the determinants
(i)$\left|\begin{array}{ccc}3& -1& -2\\ 0& 0& -1\\ 3& -5& -0\end{array}\right|$ (ii)$\left|\begin{array}{ccc}3& -4& 5\\ 1& 1& -2\\ 2& 3& 1\end{array}\right|$

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

Answer 1

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

Expanding along $R_1$

$\Delta =a_{11}{C}_{11}+a_{12}{C}_{12}+a_{13}{C}_{13}$

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

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

$=-15+3$

$\Rightarrow \Delta =-12$

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

Expanding along $R_1$

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

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

$\Rightarrow \Delta =21+20+5$

$\Rightarrow \Delta =46$

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

Expanding along $R_1$

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

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

$\Rightarrow \Delta =0+6-6$

$\Rightarrow \Delta =0$

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

Expanding along $R_1$

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

$\Delta =2(0-5)+1(0+3)-2(0-6)$

$\Rightarrow \Delta =-10+3+12$

$\Rightarrow \Delta =5$