Can AB^2009 be calculated?Matrix Multiplication and Exponentiation Limitations

[MattRSK]

Homework Statement

Matrix A and B are both square matrices of the same size.

Is it possible to compute,

AB^2009

If not, why not?

The Attempt at a Solution

Just had this question in an exam, the exam did not allow calculators and I could not answer this question, I gather it is not possible but I cannot explain why.

Because A and B are square matrices it is possible to multiply them together but I am unsure about the exponent. Any ideas?

 

[CompuChip]

Suppose that A and B are both n x n matrices. Then what is the size of AB? Then what about ABB = AB^2? And ABBB = AB^3? ...

 

[MattRSK]

Well if A and B are both n x n then they will always be n x n matrices. So does that mean that it is possible?

 

[HallsofIvy]

By the way, is that AB<sup>2009[/b] or (AB)2009?

It doesn' really matter. The only important point is that any two n by n matrices can be multiplied and give an n by n matrix as a result- which can be multiplied again by any other n by n matrix.</sup>