If A and B are arbitrary real m × n matrices, then the mapping
$\langle A, B \rangle = \text{trace}(A^T B)$
defines an inner product in $\mathbb{R}^{m \times n}$. Use this inner product to find $\langle A, B \rangle$, the norms $||A||$ and $||B||$, and the angle $\alpha_{A,B}$ between A and B for
$\begin{pmatrix} -1 & 3\\ 3 & -3 \\ 2 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & -1\\ 1 & -1 \\ 1 & -3 \end{pmatrix}$.
$\langle A, B \rangle =$
$||A|| =$
$||B|| =$
$\alpha_{A,B} = $
