Eigenvalues of $A:\;A^2 +2A=0$ 
Let $A_n$ be square matrix where $n \geq 2$ and $A^2 +2A=0$. Then
  
  
*
  
*A is singular
  
*A is nonsingular
  
*0 and -2 are eigenvalues of A
  
*either 0, or -2 is not an eigenvalue of A
  
*(1)-(4) are false.
  

My attempt: thanks to Cayley-Hamilton theorem, we know that characteristic polynomial of A equal zero. Replacing A with $\lambda$ gives $\lambda^2 + 2\lambda = 0$. Hence, 0 and -2 are eigenvalues of A, i.e. (3).
There are 2 problems: 1st, the answer should be (5), and
2nd problem is that assuming that (3) is true, (1) is also must be true due to 0 eigenvalue. So, I made mistake somewhere.

$A=0$ rules out a couple of the answers.  Can you find a non-singular matrix that satisfies $A^2+2A=0$?
