I'm a high schooler taking linear algebra this year and have two false and true questions that I'm struggling to prove.
Here are my questions,
a) If $AA^{\top}$ is singular, then so is $A$.
b) If $A+B$ is symmetric, then so are $A$ and $B$.
I tried to use $A+B=(A+B)^{\top}=A^{\top}+B^{\top}$ for the second one. but I am stuck with proving if $A+B=A^{\top}+B^{\top}$，then $A=A^{\top}$ and $B=B^{\top}$.
For (a), what are you using as the definition of singular? One way to go is to note that $\det{(AA^{\top})} = \det{(A)}\det{(A^{\top})} = \det{(A)}\det{(A)} = \left[\det{(A)}\right]^{2}$, so $\det{(AA^{\top})} = 0$ if and only if $\det{(A)} = 0$.
For (b), let $A$ be any non-symmetric matrix and let $B = -A$. Then $A+B$ is the zero matrix, which is symmetric, but $A$ and $B$ aren't symmetric.