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
Answer & explanation
Correct answer: option 4
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 $\left\{\begin{bmatrix}p&q\\q&p-q\end{bmatrix}:p,q∈R\right\}$.