# Easy to see that these two determinants are identical

1. Nov 11, 2013

### thercias

1. The problem statement, all variables and given/known data
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

2. Relevant 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.

Last edited: Nov 11, 2013
2. Nov 11, 2013

### SteamKing

Staff Emeritus
Your original determinant is somewhat garbled. You can insert text with
Code (Text):
tags to help preserve spacing.

3. Nov 11, 2013