Question

Evaluate the following:

$\left|\begin{array}{ccc}0& x{y}^2& x{z}^2\\ x^{2}y& 0& y{z}^2\\ x^{2}z& z{y}^2& 0\end{array}\right|$

Answer 1

Let $∆=\left|\begin{array}{ccc}0& x{y}^2& x{z}^2\\ x^{2}y& 0& y{z}^2\\ x^{2}z& z{y}^2& 0\end{array}\right|$.
$$\begin{gathered} ∆=\left|\begin{array}{ccc}0& x{y}^2& x{z}^2\\ x^{2}y& 0& y{z}^2\\ x^{2}z& z{y}^2& 0\end{array}\right| \\ =x^2{y}^2{z}^2\left|\begin{array}{ccc}0& x& x\\ y& 0& y\\ z& z& 0\end{array}\right|\left[\mathrm{Taking}{x}^2\mathrm{common}\mathrm{from}{C}_1,y^2\mathrm{common}\mathrm{from}{C}_2\mathrm{and}{z}^2\mathrm{common}\mathrm{from}{C}_3\right] \\ =x^3{y}^3{z}^3\left|\begin{array}{ccc}0& 1& 1\\ 1& 0& 1\\ 1& 1& 0\end{array}\right|\left[\mathrm{Taking}x\mathrm{common}\mathrm{from}{R}_1,y\mathrm{common}\mathrm{from}{R}_2\mathrm{and}z\mathrm{common}\mathrm{from}{R}_3\right] \\ =x^3{y}^3{z}^3\left|\begin{array}{ccc}0& 0& 1\\ 1& -1& 1\\ 1& 1& 0\end{array}\right|\left[\mathrm{Applying}{C}_2\to C_2-C_3\right] \\ =x^3{y}^3{z}^3\left(1+1\right)\left[\mathrm{Expanding}\mathrm{along}\mathrm{first}\mathrm{row}\right] \\ =2x^3{y}^3{z}^3 \\ \therefore ∆=2x^3{y}^3{z}^3 \end{gathered}$$