Question

Without expanding the determinant, prove that
`|{:(a+b, 2a+b, 3a+b),(2a+b, 3a+b, 4a+b), (4a+b, 5a+b, 6a+b):}|=0.`

Answer 1

Let the given determinant be `Delta`. Then, applying `C_(3) to C_(3)-C_(2),` we get:
`Delta =|{:(a+b, 2a+b, a),(2a+b, 3a+b, a), (4a+b, 5a+b, a):}|`
`=|{:(a+b, a, a),(2a+b, a, a), (4a+b, a, a):}|" "["applying"C_(2) to C_(2)-C_(1)]`
`= 0 " "[because C_(2) " and "C_(3) " are identical"]`
Hence, `Delta =0`.