Easy to see that these two determinants are identical

[thercias]

Homework Statement

Without expanding the determinant show that
bc a^2 a^2
b^2 ca b^2
c^2 c^2 ab

=
bc ab ca
ab ca bc
ca bc ab

Homework Equations

3. Attempt at solution
Well, one thing I noticed is that the diagonal row all contain the same values (bc, ca, ab)
Using the first determinant, we can simplify it to
bc |ca b^2| - ca|bc a^2| + ab |bc a^2|
...|c^2 ab|...|c^2 ab|...|b^2 ca|

the second determinant would be
bc |ca bc| - ca |bc ca| + ab|bc ab|
...|bc ab|...|ca ab|...|ab ca|obviously its easy to see that these two determinants are identical, but is this what the question asks? It says to show without expanding, so I'm not sure if there's another way to show this.

 

[SteamKing]

Your original determinant is somewhat garbled. You can insert text with

 tags to help preserve spacing.

 

[thercias]

my bad, that should be easier to read now.