Answer
(a) $$A^TA
=\left[ \begin {array}{cccc} 253&10&168&-107\\ 10&81
&-70&44\\ 168&-70&294&-139\\ -107&
44&-139&107\end {array} \right]
.$$
(b) $$A A^T=\left[ \begin {array}{ccccc} 29&30&2&86&-10\\ 30&
126&-5&169&-47\\ 2&-5&14&-37&-10
\\ 86&169&-37&425&-28\\ -10&-47&-
10&-28&141\end {array} \right]
.$$
The matrices $A^TA$ and $AA^T$ satisfy that $a_{ij}=a_{ji}$. Then, the matrices $A^TA$ and $AA^T$ are symmetric.
Work Step by Step
(a) $$A^TA=\left[ \begin {array}{ccccc} 4&2&-1&14&6\\ -3&0&-2&
-2&8\\ 2&11&0&12&-5\\ 0&-1&3&-9&4
\end {array} \right]
\left[ \begin {array}{cccc} 4&-3&2&0\\ 2&0&11&-1
\\ -1&-2&0&3\\ 14&-2&12&-9
\\ 6&8&-5&4\end {array} \right]
=\left[ \begin {array}{cccc} 253&10&168&-107\\ 10&81
&-70&44\\ 168&-70&294&-139\\ -107&
44&-139&107\end {array} \right]
.$$
(b) $$A A^T= \left[ \begin {array}{cccc} 4&-3&2&0\\ 2&0&11&-1
\\ -1&-2&0&3\\ 14&-2&12&-9
\\ 6&8&-5&4\end {array} \right]\left[ \begin {array}{ccccc} 4&2&-1&14&6\\ -3&0&-2&
-2&8\\ 2&11&0&12&-5\\ 0&-1&3&-9&4
\end {array} \right]=\left[ \begin {array}{ccccc} 29&30&2&86&-10\\ 30&
126&-5&169&-47\\ 2&-5&14&-37&-10
\\ 86&169&-37&425&-28\\ -10&-47&-
10&-28&141\end {array} \right]
.$$
The matrices $A^TA$ and $AA^T$ satisfy that $a_{ij}=a_{ji}$. Then, the matrices $A^TA$ and $AA^T$ are symmetric.
