Exercise 14

Show that each is not an equivalence on the set of $2\!\times\!2$ matrices.

• Two matrices $A,B$ are related if $a_{1,1}=-b_{1,1}$.

• Two matrices are related if the sum of their entries are within $5$, that is, $A$ is related to $B$ if $\lvert(a_{1,1}+\cdots+a_{2,2})-(b_{1,1}+\cdots+b_{2,2})\rvert<5$.