Determine Whether AA^T is Symmetric or Skew-Symmetric

To Prove:

\[ AA^T \text{ is symmetric} \]

Step 1: Take Transpose

\[ (AA^T)^T = (A^T)^T A^T \]

Step 2: Use Property

\[ (A^T)^T = A \]

\[ (AA^T)^T = AA^T \]

Conclusion:

\[ (AA^T)^T = AA^T \Rightarrow AA^T \text{ is symmetric} \]

Hence, for any square matrix, AA^T is always symmetric.