Consider the matrix A = 3 0 0

 1  5 1

-2 -4 1

(a) Show that 3 is an eigenvalue of A with algebraic multiplicity 3.

(b) Determine the geometric multiplicity of this eigenvalue.

(c) Find a basis for the eigenspace E3 = (A - 3I).

(d) Find bases for (A - 3I)^2 and (A-3I)^3 so that your basis for (A-3I)^k includes your basis for (A-3I)^(k-1). Do this by solving (A-3I)v = u and (A - 3I)w = v where u is an eigenvector in E3. Note that if your second basis already has 3 vectors, there won't be any additional vectors in the third basis.

(e) From the matrix S who columns are the three vectors in your basis for E3, and compute J = (S^-1)AS. Notice that if you order the columns of S in the most logical way, J will be triangular.

(f) Solve the system y' = Jy.

(g) Find the solution to x' = Ax by multiplying y by the appropriate matrix.