The eigenvalues of the symmetric matrix A= ſi 8 41 8 1 -4 are 11 = 9 and 12 = -9. 14 -4 7 | Find an orthogonal diagonalization of A. Find the characteristic polynomial of A.
0 ſi 1 19. (5 points) Find the eigenvalues and eigenvectors of A= 0 2 2 Lo 03 1 0 20. (5 points) Show that A= 0 2 2 is diagonalizable by finding P and D such that p-1AP = D for [003] a diagonal D.
A =10 ſi -2 -5 4 3 11 Jo 0 1 -2 0 -4 0 0 0 0 1 3 Lo 0 0 0 0 0 ] Describe all solutions of Ax = 0. x = x2 + 4
3. a) (7 pnts) Find all eigenvalues of the matrix A = 10 LO -3 6 6 3 -2 -1 11-3 b) (7 pnts) Find all eigenvectors of the matrix A = 10 lo 6 - 1 3 -2 6 c) (6 pnts) What can you say about the solution of the following system of differential equations in relation to the matrix A? Please explain briefly. X1 = x1 - 3x2 + 3x3 X2 6x2 - 2xz X3 6X2 -...
ſi -3 3 -27 3. Consider A= -3 7 -1 2 . Answer the following questions. LO 1 -4 3] a) Are the columns of A linearly independent? Justify your answer. b) Write the solution set of Ac = 0) in parametric form.
ſi -3 3 -27 3. Consider A= -3 7 -1 2 . Answer the following questions. LO 1 -4 3] a) Are the columns of A linearly independent? Justify your answer. b) Write the solution set of Ac = 0) in parametric form.
1, 5, 7, 9, 11 For each matrix A in Exercises 1 through 13, find vectors that span the kernel of A. Use paper and pencil. 1. A = 2 2. A = [1 2 3] 3. A = [08] 4. A=(: & 5. A= ſi 1 11 1 2 3 [1 3 5 6. A= [1 1 [1 2 1] 3 7. A= 8. A= ſi 231 1 3 2 13 2 si 17 1 2 1 1 [1...
3. Find all the eigenvalues and corresponding eigenspaces for the matrix B = 4. Show that the matrix B = 0 1 is not diagonalizable. 0 4] Lo 5. Let 2, and 1, be two distinct eigenvalues of a matrix A (2, # 12). Assume V1, V2 are eigenvectors of A corresponding to 11 and 22 respectively. Prove that V1, V2 are linearly independent.
4 0 2 7) Find the eigenvalues of the matrix A= 1-2 3-4 0 0 - Clearly show your work. (15 points) 3
(1 point) Find the eigenvalues of the matrix C= [7 6 1-6 3 4 -3 121 12 . -11] The eigenvalues are (Enter your answers as a comma separated list. The list you enter should have repeated items if there are eigenvalues with multiplicity greater than one.)