1.

सारणिकों के गुणधर्मों का प्रयोग करके निम्नलिखित को सिद्ध कीजिएः `|(1,x,x^(2)),(x^(2),1,x),(x,x^(2),1)|=(1-x^(3))^(2)`

Answer» L.H.S `=|(1,x,x^(2)),(x^(2),1,x),(x,x^(2),1)|=|(1+x+x^(2),x,x^(2)),(x^(2)+1+x,1,x),(x+x^(2)+1,x^(2),1)|`
`(C_(1)toC_(1)+C_(2)+C_(3))`
`=(1+x+x^(2))|(1,x,x^(2)),(1,1,x),(1,x^(2),1)|`
`=(1+x+x^(2))|(1,x,x^(2)),(0,1-x,x-x^(2)),(0,x^(2)-x,1-x^(2))|`
`(R_(2)toR_(2)-R_(1),R_(3)toR_(3)-R_(1))`
`=(1+x+x^(2))|(1,x,x^(2)),(0,1-x,x(1-x)),(0,-x(1-x),(1-x)(1+x))|`
`=(1+x+x^(2))(1-x)(1-x)|(1,x,x^(2)),(0,1,x),(0,-x,1+x)|`
`=(1+x+x^(2))(1-x)^(2).1|(1,x),(-x,1+x)|`
`=(1+x+x^(2))(1-x)^(2)(1+x+x^(2))`
`=[(1+x+x^(2))(1-x)]^(2)=(1-x^(3))^(2)`
`=R.H.S`


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