HMHD wave (U,V), (U,B)

$ \newcommand{\DS}{\displaystyle} \left[\begin{array}{rr} \DS ask+\frac{1}{ask} & \DS -\frac{1}{ask} \\ \DS -\frac{1}{ask} & \DS \phantom{\Bigg|}\frac{1}{ask} \end{array}\right] \left[\begin{array}{r} \phantom{\bigg|} U \\ \phantom{\bigg|} V \end{array}\right] = \left[\begin{array}{r} \DS \phantom{\bigg|} aW + \frac{U-V}{ask} \\ \DS \phantom{\bigg|} - \frac{U-V}{ask} \end{array}\right] = \left[\begin{array}{r} \phantom{\bigg|} aW + B \\ \phantom{\bigg|} -B \end{array}\right] = \left[\begin{array}{rr} \DS ask & \DS \phantom{\bigg|} 1 \\ 0 & \phantom{\Bigg|} -1 \end{array}\right] \left[\begin{array}{r} \phantom{\bigg|} U \\ \phantom{\bigg|} B \end{array}\right]

$ \newcommand{\DS}{\displaystyle} \left[\begin{array}{cc} ask & \phantom{\bigg|} 0 \\ \DS \frac{1}{ask} & \DS \phantom{\bigg|} -\frac{1}{ask} \end{array}\right] \left[\begin{array}{r} \phantom{\bigg|} U \\ \phantom{\bigg|} V \end{array}\right] = \left[\begin{array}{r} \phantom{\bigg|} U \\ \phantom{\bigg|} B \end{array}\right]

$ \newcommand{\DS}{\displaystyle} \left[\begin{array}{cc} 0 & \phantom{\bigg|} 1 \\ \DS 1 & \DS \phantom{\bigg|} - ask \end{array}\right] \left[\begin{array}{r} \phantom{\bigg|} U \\ \phantom{\bigg|} B \end{array}\right] = \left[\begin{array}{cc} 0 & \phantom{\bigg|} 1 \\ \DS 1 & \DS \phantom{\bigg|} - ask \end{array}\right] \left[\begin{array}{cc} ask & \phantom{\bigg|} 0 \\ \DS \frac{1}{ask} & \DS \phantom{\bigg|} -\frac{1}{ask} \end{array}\right] \left[\begin{array}{r} \phantom{\bigg|} U \\ \phantom{\bigg|} V \end{array}\right] = \left[\begin{array}{cc} \DS \frac{1}{ask} & \phantom{\bigg|} \DS -\frac{1}{ask} \\ 0 & \DS \phantom{\bigg|} 1 \end{array}\right] \left[\begin{array}{r} \phantom{\bigg|} U \\ \phantom{\bigg|} V \end{array}\right]