Chapter 1.5, Problem 56E

Set

$O=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$,

$A=\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]$, and

$B=\left[\begin{array}{cc}1& b\\ b^{-1}& 1\end{array}\right]$,

where $b\ne 0$. Show that $O$, $A$, and $B$ are solutions to the matrix equation $X^2-2X=O$. Conclude that this quadratic equation has infinitely many solutions.