If you find any mistakes, please make a comment! Thank you.

Determine which matrices belong to a given set


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 $$M A_6 =\left[ {1 \atop 1}{1 \atop 0} \right].$$

Linearity

This website is supposed to help you study Linear Algebras. Please only read these solutions after thinking about the problems carefully. Do not just copy these solutions.

Leave a Reply

Close Menu