# If the matrix $\begin{bmatrix}a&b\\c&d\end{bmatrix}$ is commutative with the matrix $\begin{bmatrix}1&1\\0&1\end{bmatrix}$ then

$\begin{array}{1 1}(a)\;a=0,b=c&(b)\;b=0,c=d\\(c)\;c=0,d=a&(d)\;d=0,a=b\end{array}$

Since matrix $\begin{bmatrix}a&b\\c&d\end{bmatrix}$ is commutative with the matrix $\begin{bmatrix}1&1\\0&1\end{bmatrix}$
$\Rightarrow \begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}1&1\\0&1\end{bmatrix}=\begin{bmatrix}1&1\\0&1\end{bmatrix}\begin{bmatrix}a&b\\c&d\end{bmatrix}$
$\Rightarrow \begin{bmatrix}a&a+b\\c&c+d\end{bmatrix}=\begin{bmatrix}a+c&b+d\\c&d\end{bmatrix}$
$\Rightarrow a=a+c,a+b=b+d$
$\Rightarrow c=0,a=d$
Hence (c) is the correct answer.