Possible Values of Determinant for Idempotent Matrix

[bcjochim07]

Homework Statement

If A is an idempotent matrix (A^2 = A), find all possible values of det(A).

Homework Equations

The Attempt at a Solution

I'm not sure if this is the proper way to show it, but here's what I did:

Since A = A^2, det(A)=det(A^2)
So det(A) = det(A)*det(A)

Considering two cases:
If det(A) is not 0, then
det(A) = 1.

The only other way to satisfy det(A) = det(A)*det(A)
is if det(A) = 0.

So det(A) is either 1 or 0.

 

[Dick]

Sure. If det(A)=x then you have x^2=x so x^2-x=0 so x*(x-1)=0. And, yes, that means x=0 or x=1.