Step 1: State the matrix and the question.
We are given $A = \begin{pmatrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{pmatrix}$ and we need to identify the nature of $AA^T$.
Step 2: Recall the definition of symmetric and skew-symmetric matrices.
A square matrix $M$ is symmetric if $M^T = M$, and skew-symmetric if $M^T = -M$.
Step 3: Take the transpose of $AA^T$.
Using the transpose property $(PQ)^T = Q^T P^T$: \[(AA^T)^T = (A^T)^T A^T = A A^T.\]
Step 4: Conclude symmetry.
Since $(AA^T)^T = AA^T$, the matrix $AA^T$ satisfies the definition of a symmetric matrix.
Step 5: Check that it is not skew-symmetric.
For $AA^T$ to be skew-symmetric, we would need $AA^T = -AA^T$, which forces $AA^T = 0$. But since $A$ is an invertible matrix (it has a nonzero determinant), $AA^T$ is a positive-definite matrix and cannot be zero. So it is definitely not skew-symmetric.
Step 6: Verify with structure argument.
For any real matrix $A$, the product $AA^T$ is always a symmetric matrix. This is a standard result in linear algebra, and this problem confirms it beautifully.
\[ \boxed{AA^T \text{ is a Symmetric Matrix}} \]