1.

तीसरे स्तम्भ के अवयवों के सहखंडों का प्रयोग करके `Delta=|(1,x,yz),(1,y,zx),(1,z,xy)|` का मान ज्ञात कीजिए .

Answer» तृतीय स्तम्भ के अवयवों का उपसारणिक है - `M_(13)=|(1,y),(1,z)|=x-y`
`M_(23)=|(1,y),(1,z)|=z-x`
`M_(33)=|(1,x),(1,y)|=y-x`
तृतीय स्तम्भ के अवयवों के सहखण्ड है -
`A_(13)=(-1)^(1+3)M_(13)=M_(13)=z-y`
`A_(23)=(-1)^(2+3)M_(23)=-M_(23)=-(z-x)=x-z`
`A_(23)=(-1)^(3+3)M_(33)=M_(33)=y-x`
`therefore Delta=a_13+a_23A_23+a_33A_33`
`=yz(z-y)+zx(x-z)+xy(y-x)" "(because a_13=yz,a_23=zx,a_33=xy)`
`=yz^2-y^2z+zx^2-z^2x+xy^2-x^2y`
`=(x^2z-x^2y)+xy^2-xz^2+yz^2+yz^2-y^2z`
`=x^2(z-y)+x(y^2-z^2)+yz(x-yz)`
`=x^2(z-y)-x(z+y)(z-y)+yz(x-y)`
`=x^2(z-y)-(z^2-y^2)+yz(z-y)`
`=(z-y)[x^2-x(z+y)+yz]`
`=(z-y)[x^2-xz-xy+yz]`
`=(z-y)[x(x-z)-y(x-z)]`
`=(z-y)(x-y)(x-z)`
`=(x-y)(z-y)(x-z)`
`=(x-y)(y-z)(z-x)` .


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