Answer
(a) $\langle A, B \rangle -5$
(b) $\| A\| =\sqrt{\langle A, A\rangle}=\sqrt{23}$
(c) $\| B \| =\sqrt{\langle B, B\rangle}=\sqrt 5$
(d) $d(A,B)=\sqrt{54}.$
Work Step by Step
$A=\left[\begin{array}{rr}{1} & {-1} \\ {2} & {4}\end{array}\right], \quad B=\left[\begin{array}{rr}{0} & {1} \\ {-2} & {0}\end{array}\right]$
(a) $\langle A, B \rangle =2a_{11} b_{11}+a_{12} b_{12}+a_{21} b_{21}+2a_{22} b_{22}=-1-4=-5$
(b) $\| A\| =\sqrt{\langle A, A\rangle}=\sqrt{2a_{11}^{2}+a_{12}^{2}+a_{21}^{2}+2a_{22}^{2}}=\sqrt{2+1+4+16}=\sqrt{23}$
(c) $\| B \| =\sqrt{\langle B, B\rangle}=\sqrt{2b_{11}^{2}+b_{12}^{2}+b_{21}^{2}+2b_{22}^{2}}=\sqrt{1+4}=\sqrt 5$
(d) $d(A,B)=\| A-B \|=\| \left[\begin{array}{rr}{1} & {-2} \\ {4} & {4}\end{array}\right] \|=\sqrt{2+4+16+32}=\sqrt{54}.$
