Determinant Functions in 3x3 Matrices for A, B, and I

[FourierX]

Homework Statement

Let A be a 3 x 3 matrix. Consider the function f(A) = a₁₁a₂₂a₃₃; g(A) = a₁₁ a₁₂ a₁₃ and h(A) =1. Show that each of these is not a determinant function.

Homework Equations

det(I) = 1
det(B) = det(A) (if B is obtained from adding a mutiple of one row of A to another row)
det(B) = - det(A) (if B is obtained from interchange two rows)
det(B) = m det(A) (if B obtained from A by multiplying a row of A by the number m)

The Attempt at a Solution

isn't f(A) = a₁₁a₂₂a₃₃ a determinant function ?

 

[naresh]

Can you show that all four conditions are satisfied for f(.)?

 

[FourierX]

what to take as a reference, though?

 

[naresh]

I don't understand what you mean by reference. You should be able to show that for any arbitrary matrix A= a_ij, if you do the things stated in the four conditions, f(.) behaves as a determinant function.

To prove that f(.) is not a determinant function, a counterexample will suffice. Trying to prove that f() is a determinant function might help you come up with this counter-example.

 

[FourierX]

I understand determinants with numbers. but its confusing with variables. Can you give me a simple example please?

 

[FourierX]

An example please...

 

[Dick]

Pick A be the identity matrix. Interchange two rows. Have you tried any numerical examples at all?