**Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 0.1 Exercise 0.1.1**

Denote by $\mathcal{A}$ the set of all $2\times 2$ 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 $\mathcal{B}$.

$$A_1 = \left[ {1 \atop 0} {1 \atop 1} \right]$$ Yes, since this matrix is simply $M$ again.

$$A_2 = \left[ {1 \atop 1} {1 \atop 1} \right]$$ No, since $$A_2 M = \left[ {1 \atop 1}{2 \atop 2} \right]$$ but $$M A_2 = \left[ {2 \atop 1}{2 \atop 1} \right].$$

$$A_3 = \left[ {0 \atop 0} {0 \atop 0} \right]$$ Yes, since anything times the zero matrix is again the zero matrix.

$$A_4 = \left[ {1 \atop 1} {1 \atop 0} \right]$$ No, since $$A_4 M = \left[ {1 \atop 1}{2 \atop 1} \right]$$ but $$M A_4 = \left[ {2 \atop 1}{1 \atop 0} \right].$$

$$A_5 = \left[ {1 \atop 0} {0 \atop 1} \right]$$ Yes, since this is the $2\times 2$ identity matrix.

$$A_6 = \left[ {0 \atop 1} {1 \atop 0} \right]$$ No, since $$A_6 M = \left[ {0 \atop 1}{1 \atop 1} \right]$$ but $$M A_6 =\left[ {1 \atop 1}{1 \atop 0} \right].$$