Find formulas for the entries of M^n, where n is a positive integer.

Question

M=[8 1]

[-2 5]

Find formulas for the entries of M^n , where n is a positive integer.

i've got [-6^n+2*7^n -2*6^n+2*7^n]

[6^n-7^n 2*6^n-7^n]

but 1x2 and 2x1 are incorrect

Answer 1

The 1x2 entry should be -6ⁿ+7ⁿ, and the 2x1 entry should be 2*6ⁿ-2*7ⁿ.

As far as the method of solution this will depend on which method you were taught in class. Here is the simplest argument that comes to mind: First you find the eigenvalues, which are 6 and 7. Next, you notice that each entry needs to be of the form a*6ⁿ+b*7ⁿ. To find these constants a and b you just need to use the entries of M⁰ (which is just the 2x2 identity matrix) and M¹ (which is the given matrix).

For example if we need to find the 2x1 entry. We write it in form a*6ⁿ+b*7ⁿ and we notice it must satisfy equations a+b=0 at n=0 and a*6+b*7=-2 at n=1. Solving the system of linear equations gives a=2, b=-2. Then we repeat the same thing for the other entries.