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  4. If | beginmatrix x y z -x y z x -y z endmatrix |=kxyz, then k =

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Q. If $ \left| \begin{matrix} x & y & z \\ -x & y & z \\ x & -y & z \\ \end{matrix} \right|=kxyz, $ then $k$ =

J & K CETJ & K CET 2007Determinants

Solution:

$ \left| \begin{matrix} x & y & z \\ -x & y & z \\ x & -y & z \\ \end{matrix} \right| $
$ =\left| \begin{matrix} x & y & z \\ -x & y & z \\ 0 & 0 & 2z \\ \end{matrix} \right| $
$ ({{R}_{3}}\to {{R}_{3}}+{{R}_{2}}) $
Expanding along $ {{R}_{3}}, $
we get $ =2z(xy+xy)=4xyz $
On comparing with kxyz, we get $ k=4 $