If $$A = \left\lbrack \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right\rbrack \ \mbox{ and } \ B = \left\lbrack \begin{array}{cc} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array} \right\rbrack$$ are arbitrary vectors in ${\mathbb R}^{2 \times 2}$, then the mapping $$\langle A,B \rangle =a_{11} b_{11} + a_{12} b_{12} + a_{21} b_{21} + a_{22} b_{22}$$ defines an inner product in ${\mathbb R}^{2 \times 2}$. Use this inner product to determine $\langle A,B\rangle$, $\|A\|$, $\|B\|$, and the angle $\alpha_{A,B}$ between $A$ and $B$ for $$A = \left[\begin{array}{cc} -3 &-3\cr 5 &2 \end{array}\right] \ \mbox{ and } \ B = \left[\begin{array}{cc} 1 &3\cr -3 &-1 \end{array}\right].$$
$\langle A,B \rangle =$ ,
$\|A\| =$ ,
$\|B\| =$ ,
$\alpha_{A,B} =$ .