Answer
(a) $$A^TA
=\left[ \begin {array}{cccc} 68&22&-6&6\\ 22&57&3&1
\\ -6&3&27&3\\ 6&1&3&10
\end {array} \right]
.$$
(b) $$A A^T=\left[ \begin {array}{cccc} 29&-14&-9&-5\\ -14&81&5
&2\\ -9&5&39&-7\\ -5&2&-7&13
\end {array} \right]
.$$
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) $$A^TA=\left[ \begin {array}{cccc} 0&8&-2&0\\ -4&4&5&0
\\ 3&0&3&-3\\ 2&1&1&2\end {array}
\right] \left[ \begin {array}{cccc} 0&-4&3&2\\ 8&4&0&1
\\ -2&5&3&1\\ 0&0&-3&2\end {array}
\right]
=\left[ \begin {array}{cccc} 68&22&-6&6\\ 22&57&3&1
\\ -6&3&27&3\\ 6&1&3&10
\end {array} \right]
.$$
(b) $$A A^T= \left[ \begin {array}{cccc} 0&-4&3&2\\ 8&4&0&1
\\ -2&5&3&1\\ 0&0&-3&2\end {array}
\right] \left[ \begin {array}{cccc} 0&8&-2&0\\ -4&4&5&0
\\ 3&0&3&-3\\ 2&1&1&2\end {array}
\right]]=\left[ \begin {array}{cccc} 29&-14&-9&-5\\ -14&81&5
&2\\ -9&5&39&-7\\ -5&2&-7&13
\end {array} \right]
.$$
The matrices $A^TA$ and $AA^T$ satisfy that $a_{ij}=a_{ji}$. Then, the matrices $A^TA$ and $AA^T$ are symmetric.