Diagonalizing a matrix?

Question

let matrix A=[ 1 1 ; 4 1]. (A 2x2 matrix with (1 1) in first row and (4 1) in the 2nd).
Diagonalize A by finding P and D such that A=PDP^-1.

Hence compute A^50.

Is it true that A^-1=(P)(D^-1)(P^-1)

please try to send me the final answers as well as the workings... thanks :)

Answer 1

The characteristic polynomial of A is p(x) = (1-x)^2 - 4. So, its eigenvalues are -1 and 3. Since they are distinct, A can be diagonalized. The matrix D = [-1 0 ; 0 3] and P is a matrix whose columns are eigenvectors of A corresponding to the eigenvalues -1 and 3.

If u = (u1, u2) is an eigenvector corresponding to -1, then A u = -u. Hence

u1 + u2 = - u1 => u2 = -2u1
4u1 + u2 = - u2 => u2 = -2u1 If we set u1 =1, then u2 = -2, so that (1, -2) is an column eigenvector of A. corresponding to the eigenvalue -1

If v is an eigenvector corresponding to 3, then

v1 + v2 = 3v1
4v1 + v2 = 3v2, so that v2 = 2v1 putting v1 =1, (1,2) is a column eigenvector corresponding to the eigenvalue 3.

It follows P = [1 1 ; -2 2] and A = P D P^(-1)

So, A^50 = P D^50 P^(-1) and D^50 = [1 0 ; 0 3^50]

Now, with an Excel spreadsheet you compute P^(-1) (or even by hand, since it's 2 X 2 ) and find A^50.