Exercise 0.1.1(Matrices which commute with $M$)

Question

In Exercises 1 to 4, let $\mathcal A$ be the set of $2 \times 2$ matrices with real number entries. Let 
$$
   M = \begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix}
$$
and let 
$$
   \mathcal B = \{ X \in \mathcal A\mid  MX = XM\}
$$
Determine which of the following elements of $\mathcal A$ lie in $\mathcal B$

$$
	\begin{pmatrix}1 & 1 \ 0 & 1\end{pmatrix},
	\begin{pmatrix}1 & 1 \ 1 & 1\end{pmatrix},
	\begin{pmatrix}0 & 0 \ 0 & 0\end{pmatrix},
	\begin{pmatrix}1 & 1 \ 1 & 0\end{pmatrix},
	\begin{pmatrix}1 & 0 \ 0 & 1\end{pmatrix},
	\begin{pmatrix}0 & 1 \ 1 & 0\end{pmatrix}
.$$

Abdullah Zubair · Accepted Answer

One can check using routine calculations that that the matrices that belong to $\mathcal B$ are 
$$
	\begin{pmatrix}1 & 1 \ 0 & 1\end{pmatrix},
	\begin{pmatrix}0 & 0 \ 0 & 0\end{pmatrix},
	\begin{pmatrix}1 & 0 \ 0 & 1\end{pmatrix}
.$$

Richard Ganaye · Answer

\begin{proof} 
Every matrix $X$ of the form $X = aI + bM\  (a,b \in \mathbb{R})$ commute with $M$:
$$(aI + bM) M = aM + bM^2 = M(aI +bM),$$
so
$ \begin{pmatrix}1 & 1 \ 0 & 1\end{pmatrix} = M = 0 I+ 1M $,  $ \begin{pmatrix}0 & 0 \ 0 & 0\end{pmatrix} = 0I + 0M$,  and $\begin{pmatrix}1 & 0 \ 0 & 1\end{pmatrix} = 1I + 0 M,$ commute with $M$.

For the three remaining matrices 
$$X_1 = \begin{pmatrix}1 & 1 \ 1 & 1\end{pmatrix},\qquad X_2 =  \begin{pmatrix}1 & 1 \ 1 & 0\end{pmatrix}, \qquad X_3 = \begin{pmatrix}0 & 1 \ 1 & 0\end{pmatrix},$$
we check if $X_i \in {\mathcal B}$:

\begin{align*}
&X_1 M = \begin{pmatrix}1 & 1 \ 1 & 1\end{pmatrix} \begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 2 \ 1 & 2 \end{pmatrix}

e
MX_1= \begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix}\begin{pmatrix}1 & 1 \ 1 & 1\end{pmatrix} = \begin{pmatrix}2& 2 \ 1 & 1\end{pmatrix},\
&X_2 M = \begin{pmatrix}1 & 1 \ 1 & 0\end{pmatrix}\begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 2 \ 1 & 1 \end{pmatrix}

e 
MX_2 = \begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix}  \begin{pmatrix}1 & 1 \ 1 & 0\end{pmatrix}= \begin{pmatrix} 2 & 1 \ 1 & 0 \end{pmatrix},\
&X_3 M = \begin{pmatrix}0 & 1 \ 1 & 0\end{pmatrix}\begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 1 \ 1 & 1 \end{pmatrix}

e 
MX_3 = \begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix}  \begin{pmatrix}0 & 1 \ 1 & 0\end{pmatrix}= \begin{pmatrix} 1 & 1 \ 1 & 0 \end{pmatrix}.\
\end{align*}
So $X_1,X_2,X_3$ are not in $B$.
\end{proof}

Exercise 0.1.1(Matrices which commute with $M$)

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Exercise 0.1.1(Matrices which commute with $M$)

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