Answer :
To get the value of m x H
We will first get the value of m and then H
To get the value of m
[tex]3\begin{pmatrix}-1 & 2 \\ 4 & 8\end{pmatrix}=\frac{2}{3}m\begin{pmatrix}-1 & 2 \\ 4 & 8\end{pmatrix}[/tex][tex]\begin{gathered} \mathrm{Switch\: sides} \\ \\ \frac{2}{3}m\begin{pmatrix}-1 & 2 \\ 4 & 8\end{pmatrix}=3\begin{pmatrix}-1 & 2 \\ 4 & 8\end{pmatrix} \end{gathered}[/tex][tex]\frac{2}{3}m\begin{pmatrix}-1 & 2 \\ 4 & 8\end{pmatrix}=\begin{pmatrix}-3 & 6 \\ 12 & 24\end{pmatrix}[/tex][tex]\begin{gathered} \frac{2}{3}m\begin{pmatrix}-1 & 2 \\ 4 & 8\end{pmatrix}=\begin{pmatrix}-3 & 6 \\ 12 & 24\end{pmatrix} \\ \\ \mathrm{Multiply\: both\: sides\: by\: }\frac{3}{2} \\ \\ \frac{3}{2}\cdot\frac{2}{3}m\begin{pmatrix}-1 & 2 \\ 4 & 8\end{pmatrix}=\frac{3}{2}\begin{pmatrix}-3 & 6 \\ 12 & 24\end{pmatrix} \\ \\ m\begin{pmatrix}-1 & 2 \\ 4 & 8\end{pmatrix}=\frac{3}{2}\begin{pmatrix}-3 & 6 \\ 12 & 24\end{pmatrix} \end{gathered}[/tex][tex]\begin{gathered} m\begin{pmatrix}-1 & 2 \\ 4 & 8\end{pmatrix}=\begin{pmatrix}-\frac{9}{2} & 9 \\ 18 & 36\end{pmatrix} \\ \mathrm{Multiply\: both\: sides\: of\: the\: equation\: by}\: \begin{pmatrix}-1 & 2 \\ 4 & 8\end{pmatrix}^{-1}\: \mathrm{from\: the\: right} \\ XA=B\quad \Rightarrow\quad \: X=BA^{-1} \\ m=\begin{pmatrix}-\frac{9}{2} & 9 \\ 18 & 36\end{pmatrix}\begin{pmatrix}-1 & 2 \\ 4 & 8\end{pmatrix}^{-1} \end{gathered}[/tex]The next is to get H
[tex](H\text{ +\lbrack{}1 4 -2\rbrack) + \lbrack{}3 2 -6\rbrack = \lbrack-2 8 -1\rbrack + (\lbrack{}1 4 -2\rbrack + \lbrack{}3 2 -6\rbrack)}[/tex]Let H be represented by [ A B C] so that
[tex]\mleft(\begin{pmatrix}A & B & C\end{pmatrix}+\begin{pmatrix}1 & 4 & -2\end{pmatrix}\mright)+\begin{pmatrix}3 & 2 & -6\end{pmatrix}=\mleft(\begin{pmatrix}-2 & 8 & -1\end{pmatrix}\mright)+\mleft(\begin{pmatrix}1 & 4 & -2\end{pmatrix}+\begin{pmatrix}3 & 2 & -6\end{pmatrix}\mright)[/tex]=>
[tex]\mleft(\begin{pmatrix}A & B & C\end{pmatrix}+\begin{pmatrix}1 & 4 & -2\end{pmatrix}\mright)+\begin{pmatrix}3 & 2 & -6\end{pmatrix}=\mleft(\begin{pmatrix}-2 & 8 & -1\end{pmatrix}\mright)+\mleft(\begin{pmatrix}1 & 4 & -2\end{pmatrix}+\begin{pmatrix}3 & 2 & -6\end{pmatrix}\mright)[/tex]=> Simplifying further
[tex]\begin{pmatrix}A+4 & B+6 & C-8\end{pmatrix}=\begin{pmatrix}2 & 14 & -9\end{pmatrix}[/tex][tex]A=-2,\: C=-1,\: B=8[/tex]Thus,
[tex]\begin{gathered} H=\lbrack A\text{ B C\rbrack} \\ H=\lbrack-2\text{ }8\text{ -1\rbrack} \end{gathered}[/tex]The final step will be to find m x H
To simplify m
Therefore
[tex]m=\frac{9}{2}[/tex]Therefore
m x H will be
[tex]\frac{9}{2}\times\lbrack-2\text{ 8 -1\rbrack}[/tex]The answer is:
[tex]\lbrack-9\text{ 36 -}\frac{9}{2}\rbrack[/tex]
