## Determine which matrices belong to a given set

Denote by $\mathcal{A}$ the set of all 2×2 matrices with real number entries. Let $M = \left[ {1 \atop 0} {1 \atop 1} \right]$ and let $\mathcal{B} = \{ X \in \mathcal{A} \ |\ MX = XM \}$.

Determine whether or not each of the following matrices is in $\mathcal{B}$.

1. $A_1 = \left[ {1 \atop 0} {1 \atop 1} \right]$: yes, since this matrix is simply $M$ again.
2. $A_2 = \left[ {1 \atop 1} {1 \atop 1} \right]$: no, since $A_2 M = \left[ {1 \atop 1}{2 \atop 2} \right]$ but $M A_2 = \left[ {2 \atop 1}{2 \atop 1} \right]$.
3. $A_3 = \left[ {0 \atop 0} {0 \atop 0} \right]$: yes, since anything times the zero matrix is again the zero matrix.
4. $A_4 = \left[ {1 \atop 1} {1 \atop 0} \right]$: no, since $A_4 M = \left[ {1 \atop 1}{2 \atop 1} \right]$ but $M A_4 = \left[ {2 \atop 1}{1 \atop 0} \right]$.
5. $A_5 = \left[ {1 \atop 0} {0 \atop 1} \right]$: yes, since this is the 2×2 identity matrix.
6. $A_6 = \left[ {0 \atop 1} {1 \atop 0} \right]$: no, since $A_6 M = \left[ {0 \atop 1}{1 \atop 1} \right]$ but $M A_6 \left[ {1 \atop 1}{1 \atop 0} \right]$.