Question

If the matrices $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ and $B=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$ commute with each other, then C = ______________.

Answer 1

It is given that, the matrices $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ and $B=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$ commute with each other.

$\therefore AB=BA$

$\Rightarrow \left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

$\Rightarrow \left[\begin{array}{cc}a& a+b\\ c& c+d\end{array}\right]=\left[\begin{array}{cc}a+c& b+d\\ c& d\end{array}\right]$

$\Rightarrow a=a+c\mathrm{and}c+d=d$

$\Rightarrow c=0$

If the matrices $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ and $B=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$ commute with each other, then c = ___0___.