Question

The determinants `|{:(1,a, bc),(1, b, ca),(1, c, ab):}| " and "|{:(1,a, a^(2)),(1, b, b^(2)),(1, c, c^(2)):}| ` are not identically equal.

Answer 1

Let `Delta = |{:(1,a, bc),(1, b, ca),(1, c, ab):}| = (1)/(abc)|{:(a,a^(2), abc),(b, b^(2), abc),(c, c^(2), abc):}| `
Applying `R_(1) to aR_(1), R_(2) to b R_(2), R_(3) to Cr_(3)`
`(1)/(abc)*abc|{:(a,a^(2), 1),(b, b^(2), 1),(c, c^(2), 1):}| = |{:(1,a, a^(2)),(1, b, b^(2)),(1, c, c^(2)):}| `
`therefore |{:(1,a, bc),(1, b, ca),(1, c, ab):}| = |{:(1,a, a^(2)),(1, b, b^(2)),(1, c, c^(2)):}|`
Hence, statement is false.