Let A = `[[2,3],[2,1]]` , D = `[[-1,0],[0,4]]`  p = `[[-1,3],[1,2]]`  a) find P^-1 and b) find a formula for A^n, where n is any positive integer. 

Expert Answers
gsenviro eNotes educator| Certified Educator

a) for a 2 x 2 matrix, A = a      b

                                       c      d

the inverse A^-1 is given as , A^-1 = 1/(ad-bc)  x d        -b

                                                                              -c        a

 here, P = -1     3

                  1     2

thus, P^-1  = 1/(-1x2 - 3x1)    2      -3

                                              -1      -1  

= 1/(-2-3)  2         -3

                -1        -1

=  1/(-5)     2        -3

                -1         -1

Thus, P^-1  = -2/5          3/5

                       1/5           1/5

b) A^n  = P D^n P^-1 

where, D is the diagonal matrix obtained by the eigenvalues of A and P is the matrix composed of eigenvectors of A. 

Hope that helps.