Answer
$$A^T=\left[\begin{array}{rrr}{1} & {3} & {-1} \end{array}\right].$$
$$A^TA=\left[\begin{array}{rrr}{11} \end{array}\right].$$
$$AA^T=\left[\begin{array}{rrr}{1} & {3} &{-1} \\ {3} & {9} &{-3}\\{-1}&{-3}&{1} \end{array}\right].$$
Work Step by Step
Let $A$ be given by
$$A=\left[\begin{array}{rrr}{1} \\ {3} \\ {-1} \end{array}\right].$$
Now, we have
$$A^T=\left[\begin{array}{rrr}{1} & {3} & {-1} \end{array}\right].$$
$$A^TA=\left[\begin{array}{rrr}{1} & {3} & {-1} \end{array}\right]\left[\begin{array}{rrr}{1} \\ {3} \\ {-1} \end{array}\right]=\left[\begin{array}{rrr}{11} \end{array}\right].$$
$$AA^T=\left[\begin{array}{rrr}{1} \\ {3} \\ {-1} \end{array}\right]\left[\begin{array}{rrr}{1} & {3} & {-1} \end{array}\right]=\left[\begin{array}{rrr}{1} & {3} &{-1} \\ {3} & {9} &{-3}\\{-1}&{-3}&{1} \end{array}\right].$$