Answer
(a) $$A^TA=\left[\begin{array}{ccc}{16} &{8}&{4} \\{8}& {8}&{0}\\{4}&{0}&{2} \end{array}\right].$$
(b) $$A A^T=\left[\begin{array}{ccc}{21} &{3} \\{3}& {5} \end{array}\right].$$
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) $$A^TA=\left[\begin{array}{ccc}{4} &{0} \\{2} & {1} \\ {1}&{-1} \end{array}\right] \left[\begin{array}{ccc}{4} &{2}&{1} \\{0}& {2} &{-1} \end{array}\right]=\left[\begin{array}{ccc}{16} &{8}&{4} \\{8}& {8}&{0}\\{4}&{0}&{2} \end{array}\right].$$
(b) $$A A^T=\left[\begin{array}{ccc}{4} &{2}&{1} \\{0}& {2} &{-1} \end{array}\right]\left[\begin{array}{ccc}{4} &{0} \\{2} & {1} \\ {1}&{-1} \end{array}\right] ]=\left[\begin{array}{ccc}{21} &{3} \\{3}& {5} \end{array}\right].$$
The matrices $A^TA$ and $AA^T$ satisfy that $a_{ij}=a_{ji}$. Then, the matrices $A^TA$ and $AA^T$ are symmetric.