Dolly Robinson

2021-01-13

Solve the equation forX, given that
$B=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$
commutes with both $\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$ and $\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]$

liannemdh

Step 1
To show that the given matrix B commutes with the other two matrices,
$B=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ commutes with $\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$ and $\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]$
To find the conditions on a, b, c, d.
For commutativity,
AB=BA
Step 2
Now, $B=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ commutes with $\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$
$\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$
$\left[\begin{array}{cc}a& 0\\ c& 0\end{array}\right]=\left[\begin{array}{cc}a& b\\ 0& 0\end{array}\right]$
So, a=a, b=c=d=0
Also, $B=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ commutes with $\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]$
$\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$
$\left[\begin{array}{cc}0& b\\ 0& d\end{array}\right]=\left[\begin{array}{cc}0& 0\\ c& d\end{array}\right]$
So, d=d, a=b=c0

Jeffrey Jordon