The determinant | * 20 c ^x C_1 & ^x C_2 & ^x C_3 ^y C_1 & ^y C_2… - Vidyadip

MCQ

The determinant $\left| {\begin{array}{*{20}{c}}{^x{C_1}}&{^x{C_2}}&{^x{C_3}}\\ {^y{C_1}}&{^y{C_2}}&{^y{C_3}}\\{^z{C_1}}&{^z{C_2}}&{^z{C_3}}\end{array}} \right|$ $=$

A
$\frac{1}{3} \,xyz (x + y) (y + z) (z + x)$
B
$\frac{1}{4} \,xyz (x + y - z) (y + z - x)$
✓
$\frac{1}{12} \, xyz (x - y) (y - z) (z - x)$
D
none

✓

Answer

Correct option: C.

$\frac{1}{12} \, xyz (x - y) (y - z) (z - x)$

c
$\left| {\,\begin{array}{*{20}{c}}x&{\frac{{x(x - 1)}}{2}}&{\frac{{x(x - 1)(x - 2)}}{6}}\\y&{\frac{{y(y - 1)}}{2}}&{\frac{{y(y - 1)(y - 2)}}{6}}\\z&{\frac{{z(z - 1)}}{2}}&{\frac{{z(z - 1)(z - 2)}}{6}}\end{array}\,} \right|$ $=$ $\frac{{xyz}}{{12}}\;\;\left| {\,\begin{array}{*{20}{c}}1&x&{{x^2}}\\1&y&{{y^2}}\\1&z&{{z^2}}\end{array}\,} \right|$

$R_1 \rightarrow R_1 - R_2 \, and \, R_2 \rightarrow R_2 - R_3$

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