Question

Using properties of determinants, prove that `|[a, a+b, a+b+c],[2a,3a+2b,4a+3b+2c],[3a,6a+3b,10a+6b+3c]|=a^3`

Answer 1

`L.H.S. = |[a,a+b,a+b+c],[2a,3a+2b,4a+3b+2c],[3a,6a+3b,10a+6b+3c]|`
Taking ,`a` common in `C_1`
`= a|[1,a+b,a+b+c],[2,3a+2b,4a+3b+2c],[3,6a+3b,10a+6b+3c]|`
Applying `R_3->R_3-R_2-R_1`
`= a|[1,a+b,a+b+c],[2,3a+2b,4a+3b+2c],[0,2a,5a+2b]|`
Applying `R_2->R_2-2R_1`
`= a|[1,a+b,a+b+c],[0,a,2a+b],[0,2a,5a+2b]|`
`=a[a(5a+2b) -(2a)(2a+b)]` `=a(5a^2+2ab-4a^2-2ab)`
`=a(a^2)`
`=a^3 = R.H.S.`