SOLUTION: Let M=[4 4 ; −2 10]. Find formulas for the entries of M^n, where n is a positive integer.

We diagnalize the matrix:



We find the eigenvalues











λ-6=0;  λ-8=0
  λ=6;    λ=8

by writing it as 



where D is the diagonal matrix with the two eigenvalues on the 
main diagonal:



and the matrix P is 



where the V's are the two column eigenvectors for the two eigenvalues

We find V1 which is the eigengvector for the eigenvalue λ=6.

We find solutions for








Divide thru by -2



We can take x1=1 and x1=1

So 



Now we do the same for the other eigenvalue

---

We find solutions for








Divide thru by -4



We can take x1=1 and x2=1

So 



So



And since the determinant of P is 1, to find P-1 we only
need to swap the elements on the the main diagonal and change the
signs of the other two elements"



Then 



So



to n factors





Any power of a diagonal matrix is the matrix whose elements are that
power of the elements, so we have the final answer as:



Edwin