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Determinants — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSDeterminants3 Marks
Question
Using cofactors of elements of third column, evaluate $\Delta=\left|\begin{array}{ccc}1 & x & y z \\ 1 & y & z x \\ 1 & z & x y\end{array}\right|$
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Answer

$\Delta = a_{13}A_{13} + a_{23}A_{23 }+ a_{33}A_{33}$
$= yz( z - y) + zx( x - z) + xy( y - x)$
$= yz^2 - y^2z + zx^2 - z^2x + xy^2 - x^2y$
$= zx^2 - x^2y + xy^2 - z^2x + yz^2 - y^2z$
$= x^2 ( z- y) + x(y^2 - z^2) + yz(z - y) $
$= (z - y)[ x^2 - x( z + y) + yz]$
$= (z - y)[x^2 - xz - xy +yz]$
$= (z - y) [ x(x - y) - z(x - y)]$
$= (z - y)[ ( x - y)(x - z)]$
$= (z - y)( x - y) (x - z)$

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