See the attached file for questioins
Thehonest
2.doc1) Determine the eigenvalues and eigenvector(s) for each matrix.
a) (8 pts)
−22 18 12 9 ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
b) (9 pts)
1 0 0 2 2 0 3 3 3
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥
2) (8 pts) The trace of a square matrix is the sum of its diagonal entries. This is written tr(A) = ∑aii. Square matrices have a neat property where the trace of the matrix is equal to the sum of its eigenvalues. Also, the determinant of a matrix is equal to the product of its eigenvalues. Written in math speak:
tr(A) = ∑λi
det(A) = ∏λi
Use the trace and the determinant properties to get a system of two (nonlinear) equations in λ1 and λ2 and solve for those eigenvalues for the matrix
C = 5 10
−2 −3 ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
3) Give all of the eigenspaces for each matrix. Give the algebraic and geometric multiplicity associated with each eigenvalue.
b) (15 pts)
5 1 0 0 0 5 0 0 0 0 5 0 0 0 0 3
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥
4) (5 pts) Compute PD2P-1 and A2. What do you notice?
5) (5 pts) Compute
PD 3P−1
A3 = 13 7
28 13 ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
6) (10 pts) Give a general equation to compute Ak. This works for every diagonalizable matrix A. Why does this work? Recall that A = PDP-1.
