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].
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Comments

  • Nicolas  On September 20, 2013 at 8:11 pm

    Muchas gracias por el tiempo de escribir esto y explicarlo tan claro.

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