I'm reviewing for my linear algebra course, and have four "true or false" questions that I'm struggling to prove. I've included my approach to the solutions in brackets below them:

1) **If $A^2 = B^2$, then A = B or A = -B, where A and B are nxn matrices**   

(Not sure how to approach this one at all)

2) **Every 3x3 skew symmetric matrix is singular**   

(Pretty sure I have this one correct: Because this is a skew symmetric matrix, $\det(A) = \det(A^T) = \det(-A) = (-1)^n\det(A)$, and when n is odd $\det(A) = -\det(A)$, so $2\det(A) = 0$ and therefore $\det(A) = 0$. As such, the answer is "False" because it is only singular when n is odd)

3) **Any system of n linear equations in n variables has at most n solutions**   

(A system can have infinitely many solutions if the determinant is zero, right? I just don't know how to prove it)

4) **For a square matrix A, A is invertible if and only if $AA^T$ is**   

(Not sure how to approach this one, either)