The set of all 2 × 2 matrices which commute with the matrix $\begin{bmatrix}1&1\\1&0\end{bmatrix}$ with respect to matrix multiplication is

$\left\{\begin{bmatrix}p&q\\q&p-q\end{bmatrix}:p,q∈R\right\}$

Let $A = \begin{bmatrix}p&q\\r&s\end{bmatrix}$ be a matrix which commute with matrix $B =\begin{bmatrix}1&1\\1&0\end{bmatrix}$. Then,

$AB=BA$.

$\begin{bmatrix}p&q\\r&s\end{bmatrix}\begin{bmatrix}1&1\\1&0\end{bmatrix}=\begin{bmatrix}1&1\\1&0\end{bmatrix}\begin{bmatrix}p&q\\r&s\end{bmatrix}$

$⇒\begin{bmatrix}p+q&p\\r+s&r\end{bmatrix}=\begin{bmatrix}p+r&q+s\\p&q\end{bmatrix}$

$⇒p+q=p+r,p=q+s,r+s=p$ and, $r = q$

$⇒r = q$ and $s=p-q$

$∴A=\begin{bmatrix}p&q\\q&p-q\end{bmatrix}$

Hence, required set is $\left\{\begin{bmatrix}p&q\\q&p-q\end{bmatrix}:p,q∈R\right\}$.