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Using properties of determinants, prove that : |[1+a,1,1],[1,1+b,1],[1,1,1+c]|=abc + bc + ca + ab - Mathematics

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Using properties of determinants, prove that :

`|[1+a,1,1],[1,1+b,1],[1,1,1+c]|=abc + bc + ca + ab`

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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`

Concept: Elementary Transformations
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