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Prove that : Det[[1+a,1,1],[1,1+b,1],[1...

Prove that : `Det[[1+a,1,1],[1,1+b,1],[1,1,1+c]]=abc(1+1/a+1/b+1/c)`

Text Solution

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LHS= `|(1+a, 1,1),(1, 1+b, 1),(1,1,1+c)|`
`R_1-> R_1-(1+a)R_3`
`R_2-> R_2 - R_3`
`|(0,-a,1-(1+c)(1+a)),(0,b,-c),(1,1,1+c)|`
`= |(0,-a,-c-a+ac),(0,b,-c),(1,1,1+c)|`
expand `c_1`
`= 0+0+1|(-a,-c-a+ac),(b,-c)| `
`= ac -[(b)(-c-a-ac)]`
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