Question

Using properties of determinants, prove that :

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

Answer 1

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`