Question

सारणियों के गुणों का प्रयोग कर दिखाएँ कि
`|{:(b^(2)c^(2),bc,b+c),(c^(2)a^(2),ca,c+a),(a^(2)b^(2),ab,a+b):}|=0`

Answer 1

माना कि `Delta=|{:(b_(2)c^(2),bc,b+c),(c^(2)a^(2),ca,c+a),(a^(2)b^(2),ab,a+b):}|`
`R_(1)` से bc तथा `R_(2)` से ca तथा `R_(3)` से ab, common लेने पर
`Delta=a^(2)b^(2)c^(2)|{:(bc,a,(1)/(b)+(1)/(c)),(ca,1,(1)/(c)+(1)/(a)),(ab,a,(1)/(2)+(1)/(b)):}|=a^(2)b^(2)c^(2)|{:(bc,1,(1)/(b)+(1)/(c)),(c(a-b),0,(1)/(a)-(1)/(b)),(b(a-c),0,(1)/(a)-(1)/(c)):}|" "[R_(2)toR_(2)-R_(1),R_(3)toR_(3)-R_(1)]`
`=a^(2)b^(2)c^(2)(a-b)(a-c)|{:(bc,a,(1)/(b)+(1)/c),(c,0,-(1)/(ab)),(b,0,-(1)/(ac)):}|=(a^(2)b^(2)c^(2)(a-b)(a-c))/(ab*ac)|{:(bc,a,(1)/(b)+(1)/(c)),(abc,0,-1),(abc,0,-1):}|" "[R_(2)"में ab तथा "R_(3) "में से गुणा करने पर"] `
`=bc(a-b)(a-c)xx0=0`.