1.

साबित करे 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 कि `Delta=|{:(1+a,1,1),(1,1+b,1),(1,1,1+c):}|` तो, `Delta=(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 common 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)" का प्रयोग करने तथा " R_(1) "से "((1)/(a)+(1)/(b)+(1)/(c)+1)" common लेने पर"]`
`=(abc)((1)/(a)+(1)/(b)+(1)/(c)+1)|{:(0,0,1),(0,1,(1)/(b)),(-1,-1,(1)/(c)+1):}|" "[C_(1)to(C_(1)-C_(3))" तथा "C_(2)to(C_(2)-C_(3))]`
`=(abc)((1)/(a)+(1)/(b)+(1)/(c)+1)*(1)*|{:(0,1),(-1,-1):}|" "[R_(1)"के अनुदिश विस्तार करने पर"]`
`=(abc)((1)/(a)+(1)/(b)+(1)/(c)+1)*1=(abc)((1)/(a)+(1)/(b)+(1)/(c)+1)=(bc+ca+ab+abc)`


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