Answer
$Ab=BA$ if and only if $x=-y$ and $w=z$.
Work Step by Step
Since we have
$$AB=\left[\begin{array}{cc}{w} &{x} \\{y}& {z} \end{array}\right] \left[\begin{array}{cc}{1} &{1} \\{-1}& {1} \end{array}\right]=\left[\begin{array}{cc}{w-x} &{w+x} \\{y-z}& {y+z} \end{array}\right],$$
$$BA=\left[\begin{array}{cc}{1} &{1} \\{-1}& {1} \end{array}\right] \left[\begin{array}{cc}{w} &{x} \\{y}& {z} \end{array}\right] =\left[\begin{array}{cc}{w+y} &{x+z} \\{y-w}& {z-x} \end{array}\right].$$
Then, $Ab=BA$ if and only if $x=-y$ and $w=z$.
