Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies.
We are not going to multiply A with itself a hundred times, at least not directly. Instead, we are going to diagonalize A first, because the 100th power of a diagonal matrix is much easier to calculate. Indeed, if D is the diagonal matrix corresponding to A (i.e. is a matrix with the eigenvalues of A on the diagonal), and if T is the transformation matrix of eigenvectors of A, then
So let's find D, T and T -1. We need the eigenvalues and eigenvectors of A, so we calculate the roots of the characteristic polynomial :
And so the eigenvalues are 1/2, -1/3 and 1.
We compute an eigenvector for each eigenvalue using row reduction on the system .
For λ = 1 we have the system
And so this eigenspace is spanned by the eigenvector [2, 3, 3].
For λ = 1/2 we have the system
And so this eigenspace is spanned by the eigenvector [1, -1, 0].
For λ = -1/3 we have the system
And so this eigenspace is spanned by the eigenvector [2, 3, -5].
Using this basis of eigenvectors, we have that
We need now to compute the inverse of the matrix T. We obtain
So we can compute x100
As n becomes very large, xn approaches