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Question
Using properties of determinants, prove that :
`|[1+a,1,1],[1,1+b,1],[1,1,1+c]|=abc + bc + ca + ab`
Solution
Consider the detrminant
`Delta=|[1+a,1,1],[1,1+b,1],[1,1,1+c]|`
Taking abc common outside, we have
`Delta=abc|[1/a+a,1/b,1/c],[1/a,1/b+1,1/c],[1/a,1/b,1/c+1]|`
Apply the transformation, C1→ C1+C2+C3
`Delta=abc|[1+1/a+1/b+1/c,1/b,1/c],[1+1/a+1/b+1/c,1/b+1,1/c],[1+1/a+1/b+1/c,1/b,1/c+1]|`
`=>Delta=abc(1+1/a+1/b+1/c)|[1,1/b,1/c],[1,1/b+1,1/c],[1,1/b,1/c+1]|`
Apply the transformations R2→ R2-R3 and R3→ R3-R1
`Delta=abc(1+1/a+1/b+1/c)|[1,1/b,1/c],[0,1,0],[0,0,1]|`
Expanding along C1 , we have
`Delta=abc(1+1/a+1/b+1/c)xx1xx|[1,0],[0,1]|`
`Delta=abc(1+1/a+1/b+1/c)=abc+ab+bc+ca`
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