Proving Simple Linear Algebra Statement: A^2 = A implies A is either 0 or I

[CuppoJava]

Homework Statement

Prove that given a matrix A, and A^2 = A, then A must be either the zero matrix or the identity matrix.

The Attempt at a Solution

By multiplying both sides by A, you can deduce that A = A^2 = A^3 = A^4 ...
From there I think it's obvious that A must be either 0 or I, but I don't know how to start proving it formally.

Thanks very much for your help
-Patrick

 

[VeeEight]

Try taking the determinant of A

 

[CuppoJava]

Is there a way to prove it without using the determinant. This exercise is given before determinants are introduced.

Thanks
-Patrick

 

[HallsofIvy]

If $A^2= A$, then $A^2- A= A(A- I)= 0$.

However, you have to be careful here. With matrices it is NOT true that "if AB= 0 then either A= 0 or B= 0".

 

[buzzmath]

Is that true? What about A = [1,0;0,0]? This is not zero or I but A^2 = A

 

[CuppoJava]

Ah. I didn't spot that buzzmath. Thank you. The question actually does say either prove or find a counterexample. I was just too sure that it was true.

Thanks
-Patrick