Algebra Solutions: 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 $\mathcal{B}$ .

$A_1 = \left[ {1 \atop 0} {1 \atop 1} \right]$ : 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 2×2 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 $M A_6 \left[ {1 \atop 1}{1 \atop 0} \right]$ .

Algebra Solutions

Determine which matrices belong to a given set

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