Question

If $A=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$ and $B=\dots$ then $AB=BA$ where $B\ne I$.

• A. $\left[\begin{array}{cc}x& 0\\ y& y\end{array}\right]$
• B. $\left[\begin{array}{cc}x& y\\ 0& x\end{array}\right]$
• C. $\left[\begin{array}{cc}x& y\\ 0& y\end{array}\right]$
• D. $\left[\begin{array}{cc}x& x\\ y& 0\end{array}\right]$

Answer 1

Answer: (B)

$A=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$ let $B=\left[\begin{array}{cc}x& z\\ y& w\end{array}\right]$
$AB=BA$
$\Rightarrow \left[\begin{array}{cc}x+y& z+w\\ y& w\end{array}\right]=\left[\begin{array}{cc}llx& x+z\\ y& w+y\end{array}\right]$
$x+y=x\Rightarrow y=0$
$x=w$
$\Rightarrow B=\left[\begin{array}{cc}x& z\\ 0& x\end{array}\right]$