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Let M be the set of all 3x3 non-singular matrices. Define a relation R on M x M by R = {(A, B): A*B = B*A}. Which of the following correctly describes R?

1. Reflexive, symmetric but not transitive
2. Reflexive, symmetric and transitive
3. Reflexive, transitive but not symmetric
4. Neither reflexive nor symmetric nor transitive

Correct answer: Reflexive, symmetric but not transitive

Solution

R is reflexive (AA=AA for all A) and symmetric (AB=BA implies BA=AB). It is not transitive: for example, if B = I (identity), then any A commutes with I and any C commutes with I, but A need not commute with C.

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