Answer
(a) $\langle A, B \rangle -6$
(b) $\| A\| =\sqrt{35}$
(c) $\| B \| =\sqrt{7}$
(d) $d(A,B)=\| A-B \| =\sqrt{54}.$
Work Step by Step
$A=\left[\begin{array}{rr}{-1} & {3} \\ {4} & {-2}\end{array}\right], \quad B=\left[\begin{array}{rr}{0} & {-2} \\ {1} & {1}\end{array}\right]$
(a) $\langle A, B \rangle =2a_{11} b_{11}+a_{12} b_{12}+a_{21} b_{21}+2a_{22} b_{22}=-6+4-4=-6$
(b) $\| A\| =\sqrt{\langle A, A\rangle}=\sqrt{a_{11}^{2}+a_{12}^{2}+a_{21}^{2}+2a_{22}^{2}}=\sqrt{2+9+16+8}=\sqrt{35}$
(c) $\| B \| =\sqrt{\langle B, B\rangle}=\sqrt{2b_{11}^{2}+b_{12}^{2}+b_{21}^{2}+2b_{22}^{2}}=\sqrt{0+4+1+2}=\sqrt{7}$
(d) $d(A,B)=\| A-B \|=\| \left[\begin{array}{rr}{-1} & {5} \\ {3} & {-3}\end{array}\right] \|=\sqrt{2+25+9+18}=\sqrt{54}.$
