Exercise 10

Consider the following relationship on the set of $2\!\times\!2$ matrices: we say that $A$ is sum-what like $B$ if the sum of all of the entries in $A$ is the same as the sum of all the entries in $B$. For instance, the zero matrix would be sum-what like the matrix whose first row had two sevens, and whose second row had two negative sevens. Prove or disprove that this is an equivalence relation on the set of $2\!\times\!2$ matrices.