# If the matrices

Question:

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

Solution:

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

$\therefore A B=B A$

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

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

$\Rightarrow a=a+c$ and $c+d=d$

$\Rightarrow c=0$

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