Easy to see that these two determinants are identical

In summary, the conversation discusses how to show that two determinants are equal without expanding them. The solution involves simplifying the first determinant by noticing that its diagonal row contains the same values, and then comparing it to the second determinant to show that they are identical. The question asks if there is another way to show this without expanding the determinants.
  • #1
thercias
62
0

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.
 
Last edited:
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  • #2
Your original determinant is somewhat garbled. You can insert text with
Code: 
 tags to help preserve spacing.
 
  • #3
my bad, that should be easier to read now.
 

What does it mean when two determinants are identical?

When two determinants are identical, it means that they have the same values for each corresponding element. This means that they have the same number of rows and columns, and each element in the same position has the same value.

How can you tell if two determinants are identical?

You can tell if two determinants are identical by comparing the values of each corresponding element. If they have the same values for each element, then they are identical. Another way is to check if they have the same number of rows and columns.

Why is it important to know if two determinants are identical?

Knowing if two determinants are identical is important because it can help simplify calculations and solve problems more efficiently. Identical determinants have the same properties and can be used interchangeably in certain operations.

What are some properties of identical determinants?

Some properties of identical determinants include having the same values for each corresponding element, the same number of rows and columns, and the same determinant value. They also have the same rank and can be reduced to the same triangular form.

Can two determinants be identical if they have different dimensions?

No, two determinants cannot be identical if they have different dimensions. Identical determinants must have the same number of rows and columns and the same values for each corresponding element. If they have different dimensions, they are not considered identical.

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