EXAMPLE 19 Prove that y +z zy Z+X X...
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EXAMPLE 19 Prove that y +z zy Z+X X y x x+y = 4xyz.

11th - 12th Class
Maths
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65
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( begin{array}{l}text { SOLUTION Given determinant }=left|begin{array}{ccc}y+z & z & y z & z+x & x y & x & x+yend{array}right| =left|begin{array}{ccc}0 & -2 x & -2 x z & z+x & x y & x & x+yend{array}right| & {left[R_{1} rightarrow R_{1}-left(R_{2}+R_{3}right)right]} & =(-2 x) cdotleft|begin{array}{cc}0 & 1 & 1 y & z+x & x y & x & x+yend{array}right|end{array} ) ( left[operatorname{taking}(-2 x) operatorname{common} mathrm{from} mathrm{R}_{1}right. ) ( =(-2 x) cdotleft|begin{array}{ccc}0 & 0 & 1 z & z & x y & -y & x+yend{array}right| quadleft[C_{2} rightarrowleft(C_{2}-C_{3}right)right] ) ( =(-2 x) cdot 1 cdotleft|begin{array}{lr}z & z y & -yend{array}right| quadleft[text { expanded by } R_{1}right] ) ( =(-2 x) cdot 1 cdot(-y z-y z)=(-2 x)(-2 y z)=4 x y z )
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