If `a ,b` and `c` are all non-zero and `|1+a1 1 11+b1 1 11+c|=0,` then prove that `1/a+1/b+1/c+1=0`

Explanation

Given: `|(1+a,1, 1),( 1,1+b,1),( 1, 1,1+c)|=0,`

Then we have to proof that:`1/a+1/b+1/c+1=0`

Let `|(1+a,1, 1),( 1,1+b,1),( 1, 1,1+c)|=0,`

Now apply `C_1->C_1-C_2`

`|(a,1, 1),( -b,1+b,1),( 0, 1,1+c)|=0,`

Now apply `C_2->C_2-C_3`

`|(a,0, 1),( -b,b,1),( 0, -c,1+c)|=0,`

`a(b+bc+c)+1(bc)=0`

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