Solution

Recall:

• If $A$ is $m\times n$, then $AA^T$ is $m\times m$ and $A^TA$ is $n\times n$.

(i) $A=\begin{bmatrix}-3&2&-1\\2&1&3\end{bmatrix}$

First compute $A^T$:

Compute $AA^T$

Compute $A^TA$

Compute $|A^TA|$ (expansion gives $0$):

So:

(ii) $A=\begin{bmatrix}3&1\\2&2\\1&3\end{bmatrix}$

Then:

Compute $AA^T$

This matrix has determinant $0$:

Compute $A^TA$

Therefore: