1.

सिद्ध करो कि `|{:(1+a,1,1),(1,1+b,1),(1,1,1+c):}|=(abc)((1)/(a)+(1)/(b)+(1)/(c)+1)` `=(bc+ca+ab+abc)`

Answer» माना `Delta=|{:(1+a,1,1),(1,1+b,1),(1,1,1+c):}|`
`=abc|{:((1)/(a)+1,(1)/(a),(1)/(a)),((1)/(b),(1)/(b)+1,(1)/(b)),((1)/(c),(1)/(c),(1)/(c)+1):}|`
`(R_(1),R_(2)` तथा `R_(3)` से क्रमशः a,b,c उभयनिष्ठ लेने पर)
`=(abc)((1)/(a)+(1)/(b)+(1)/(c)+1)|{:(1,1,1),((1)/(b),(1)/(b)+1,(1)/(b)),((1)/(c),(1)/(c),(1)/(c)+1):}|`
`(R_(1)toR_(1)+R_(2)+R_(3)` क्रिया द्वारा तथा `R_(1)` से `((1)/(a)+(1)/(b)=(1)/(c)+1)` उभयनिष्ठ लेने पर)
`=(abc)((1)/(a)+(1)/(b)+(1)/(c)+1)|{:(0,0,1),(0,1,(1)/(b)),(-1,-1,(1)/(c)+1):}|(C_(1)toC_(1)-C_(3)"तथा"C_(2)toC_(2)-C_(3))`
`=(abc)((1)/(a)+(1)/(b)+(1)/(c)+1)1`
`=(abc)((1)/(a)+(1)/(b)+(1)/(c)+1)`
`=(ab+bc+ca+abc)`


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