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Prove |(1+a,1,1),(1,1+b,1),(1,1,1+c)|=ab...

Prove `|(1+a,1,1),(1,1+b,1),(1,1,1+c)|=abc(1/a+1/b+1/c+1)`

Text Solution

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`L.H.S. = |[1+a,1,1],[1,1+b,1],[1,1,1+c]|`
Applying `C_1->C_1-C_2`
`=>|[a,1,1],[-b,1+b,1],[0,1,1+c]|`
Applying `C_2->C_2-C_3`
`=>|[a,0,1],[-b,b,1],[0,-c,1+c]|`
`=>[a(b+bc+c)+1(bc)]`
`=> ab+abc+ac+bc`
`=>abc+bc+ac+ab`
...
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