Question

Evaluate $\left|\begin{array}{ccc}{}^x{C}_1& {}^x{C}_2& {}^x{C}_3\\ {}^y{C}_1& {}^y{C}_2& {}^y{C}_3\\ {}^z{C}_1& {}^z{C}_2& {}^z{C}_3\end{array}\right|.$

Answer 1

$\Delta =\left|\begin{array}{ccc}X& X(X-1)/1.2& X(X-1)(X-2)/\mathrm{1.2.3}\\ Y& Y(Y-1)/1.2& Y(Y-1)(Y-2)/\mathrm{1.2.3}\\ Z& Z(Z-1)/1.2& Z(Z-1)(Z-2)/\mathrm{1.2.3}\end{array}\right|$

$=\frac{XYZ}{2.6}\left|\begin{array}{ccc}1& (X-1)& (X-1)(X-2)\\ 1& (Y-1)& (Y-1)(Y-2)\\ 1& (Z-1)& (Z-1)(Z-2)\end{array}\right|$

[Operate $R_2-R_1$ and $R_3-R_1$ and expend with $C_1$]

$=\frac{XYZ}{12}\left|\begin{array}{ccc}1& (X-1)& (X-1)(X-2)\\ 0& (Y-X)& (Y-X)(Y+X-3)\\ 0& (Z-X)& (Z-X)(Z+X-3)\end{array}\right|$

$=\frac{XYZ}{12}.(Y-X)(Z-X)\left|\begin{array}{cc}1& Y+X-3\\ 1& Z+X-3\end{array}\right|$

$=\frac{XYZ}{12}(Y-X)(Z-X)[(Z+X-3)-(Y+X-3)]$

$=\frac{XYZ}{12}(Y-X)(Z-X)(Z-Y)$

$=\frac{1}{12}XYZ(Y-Z)(Z-X)(X-Y)$.