About Singular and Symmetric Matrix

[jack1234]

I would like to know the statement is always true or sometimes false, and what is the reason:
A is a square matrix
P/S: I denote transpose A as A^T
1)If AA^T is singular, then so is A;
2)If A^2 is symmetric, then so is A.

 

[StatusX]

Is this homework? What have you tried?

 

[jack1234]

This is not homework, I just want to practise more.
About the question, I have no hint whether the statement is correct or false.
Can you give me some hint so i can start solving this problem?

 

[StatusX]

Ok, if AB is singular, can you say anything about A or B? Also, when checking if a matrix is symmetric, the natural thing to do is look at its transpose.

EDIT: Sorry, I read your second question backwards (ie, to show A^2 is symmetric if A is). As a hint for your actual question, note that the zero matrix is symmetric.