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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...?????