Prove that `|{:(a,,a^(2),,bc),(b ,,b^(2),,ac),( c,,c^(2),,ab):}|`=`|{:(1,,1,,1),(a^(2) ,,b^(2),,c^(2)),( a^(3),, b^(3),,c^(3)):}|`

Explanation

we have `Delta =|{:(a,,a^(2),,bc),(b,,b^(2),,ac),(c^(2),,c^(3),,ab):}|`
Multiplying `R_(1),R_(2) " and " R_(3) by a, b," and " c ` respectively we get
`Delta =(1)/(abc) |{:(a^(2),,a^(3),,abc),(b^(2),,b^(3),,abc),(c^(2),,c^(3),,abc):}|`
`=|{:(a^(2),,a^(3),,1),(b^(2),,b^(3),,1),(c^(2),,c^(3),,1):}|` (Taking abc common from `C_(3))`
`=- |{:(1,,a^(3),,a^(2)),(1,,b^(3),,b^(2)),(1,,c^(3),,c^(2)):}|" "(C_(1) hArr C_(3))`
`=|{:(1,,a^(2),,a^(3)),(1,,b^(2),,b^(3)),(1,,c^(2),,c^(3)):}|" "(C_(2) hArr C_(2))`
`=|{:(1,,1,,1),(a^(2),,b^(2),,c^(2)),(a^(3),,b^(3),,c^(3)):}| " " "(Taking transpose)"`