Determinant of row interchange proof

[fackert]

Need a lot of help here guys.

I need to prove that for an nxn matrix A, if i interchange two rows to obtain B, then det=-detA

I have proved my basis (below), but I'm stuck on the hard part, the induction (which I'm required to do). I understand the steps of induction, but i don't know how to do it for this case.

What i have so far:

Let A be an nxn matrix.
Basis n=2
Then detA=a(11)a(22) - a(12)a(21)
Now let B be the matrix obtained by interchanging rows 1 and 2
Then detB=a(21)a(12) - a(22)a(11)
=-detA
So true for an arbitary 2x2 matrix.

(induction)
Assume true for n=k
For a kxk matrix...?

 

[chiro]

Hey fackert.

Try introducing a pre-multiplication matrix that swaps the rows and then use the relationship that det(A*B) = det(A)*det(B) where B is your original matrix and A is a matrix transformation that swaps the rows.

 

[mathwonk]

my notes on determinants are on page 62ff of these notes:

http://www.math.uga.edu/%7Eroy/4050sum08.pdf