If $A$ and $B$ are arbitrary real $m\times n$ matrices, then the mapping $$\langle A,B \rangle = {\rm trace}(A^T B)$$ 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 $$A = \left[\begin{array}{cc} -2 &1\cr -2 &2\cr 3 &-2 \end{array}\right] \ \mbox{ and } \ B = \left[\begin{array}{cc} 2 &-1\cr 1 &-1\cr 2 &2 \end{array}\right].$$
$\langle A,B \rangle =$ ,
$\|A\|=$ ,
$\|B\|=$ ,
$\alpha_{A,B}=$ .