SOLUTION: Multiplication of matrices are only commutative for some cases of matrices. What are these cases?

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Question 102267: Multiplication of matrices are only commutative for some cases of matrices. What are these cases?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Off the top of my head I can only think of these cases where commutativity holds:


I%2AA=A%2AI where I is the identity matrix


A%5E-1%2AA=A%2AA%5E-1 where A%5E-1 is the inverse of matrix A


Also, according to Wolfram Mathworld, "...matrix multiplication is not, in general, commutative (although it is commutative if A and B are diagonal and of the same dimension)."



For instance, if you have matrices diagonal matrices A and B


A=%28matrix%283%2C3%2C2%2C0%2C0%2C0%2C3%2C0%2C0%2C0%2C5%29%29, B=%28matrix%283%2C3%2C5%2C0%2C0%2C0%2C6%2C0%2C0%2C0%2C7%29%29


the first product AB is
AB=%28matrix%283%2C3%2C10%2C0%2C0%2C0%2C18%2C0%2C0%2C0%2C35%29%29



and the second product BA is
BA=%28matrix%283%2C3%2C10%2C0%2C0%2C0%2C18%2C0%2C0%2C0%2C35%29%29


which is the same product as AB. So if you were to do this with general entries of the matrices A and B, you would find that AB=BA only if A and B are diagonal matrices and they are both the same size