What is m to the power of n?

Question

Thad just learned about matrix multiplication. He began to wonder what happens when you take powers of a matrix. He computed the first few powers of the matrix m= 1 1 and noticed a pattern. What is m to the power of n?
                                                                      0 1

Answer 1

You can take the power of a matrix only if it is a square matrix.  The problem statement says "m = 1 1".  That looks like a row.  Is "0 1" the second row of a 2x2 matrix?  Is the matrix

 

m = ⌈1 1⌉ ?

       ⌊0 1⌋

 

If so, then

 

m² = ⌈1 2⌉

         ⌊0 1⌋

    

m³ = ⌈1 3⌉

         ⌊0 1⌋

 

m⁴ = ⌈1 4⌉

         ⌊0 1⌋

 

It looks like you get the pattern:

 

mⁿ = ⌈1 n⌉

         ⌊0 1⌋