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If `a ,b` and `c` are all non-zero and `|[1+a,1, 1],[ 1,1+b,1],[1,1,1+c]|=0,` then prove that `1/a+1/b+1/c+1=0` |
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Answer» `|[1+a,1,1],[1,1+b,1],[1,1,1+c]| = 0` Applying `C_1->C_1-C_2` `=>|[a,1,1],[-b,1+b,1],[0,1,1+c]| = 0` Applying `C_2->C_2-C_3` `=>|[a,0,1],[-b,b,1],[0,-c,1+c]| = 0` `=>[a(b+bc+c)+1(bc)] = 0` `=> ab+abc+ac+bc = 0` `=>abc+bc+ac+ab = 0` Dividing by `abc` on both sides, `=>1+1/a+1/b+1/c = 0` |
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