Aaron Lauve Math315-488-fall-2012 Assignment Hw05 due 10/24 ...

Aaron LauveAssignment Hw05 due 10/24/2012 at 11:00pm CDTMath315-488-fall-2012/ur la 11 22a.pg ⎡⎤•1 1 2 -1•The matrix A = ⎢ 1 1 2 -1⎥⎣ -1 -1 -2 1 ⎦ . •(incorrect)has two real eigenvalues λ 1 < λ 2 . Find these eigenvalues,1. (1 pt) Library/Rochester/setLinearAlgebra13ComplexEigenvalues-their multiplicities, and dimensions of the corresponding/ur la 13 6.pg [ ]eigenspaces.-1 -3Let M = .λ 1 = has multiplicity . The dimension of the correspondingeigenspace is .3 -1Find formulas for the entries of M n where n is a positive integer. λ 2 = has multiplicity . The dimension of the correspondingeigenspace is .(Your[ formulas should not contain complex ] numbers.)M n =.Is the matrix A defective? (Type ”yes” or ”no”) .Answer(s) submitted:Answer(s) submitted:•••••••••(incorrect)••2. (1 pt) Library/TCNJ/TCNJ Diagonalization/problem2.pgA, P and D are n × n matrices.Check the true statements below:(incorrect)5. (1 pt) Library/Rochester/setLinearAlgebra11Eigenvalues-/ur la 11 11.pg[ Find ] a 2 × 2[ matrix ] A such that• A. If A is diagonalizable, then A has n distinct eigenvalues.0 14 2and• B. If AP = PD, with D diagonal, then the nonzero are eigenvectors [ of A, with ] eigenvalues 6 and −4 respectively.columns of P must be eigenvectors of A.A =.• C. A is diagonalizable if A has n distinct eigenvectors.• D. If A is invertible, then A is diagonalizable.Answer(s) submitted:Answer(s) submitted:•(incorrect)••••3. (1 pt) Library/Rochester/setLinearAlgebra11Eigenvalues- (incorrect)/ur la 11 17.pg [ ]-8 26. (1 pt) Library/Rochester/setLinearAlgebra11Eigenvalues-The matrix A =/ur la 11 9.pg-2 -4has one eigenvalue of multiplicity 2. Find this eigenvalue and Supppose A is an invertible n × n matrix and v is an eigenvectorthe dimenstion of the eigenspace.of A with associated eigenvalue 3. Convince yourself that v iseigenvalue = ,an eigenvector of the following matrices, and find the associateddimension of the eigenspace = .eigenvalues:Answer(s) submitted:1. A 3 , eigenvalue = ,2. A•, eigenvalue = ,•3. A − 2I n , eigenvalue = ,4. 7A, eigenvalue = .(incorrect)Answer(s) submitted:4. (1 pt) Library/Rochester/setLinearAlgebra11Eigenvalues- •1 1 2 -11

7. (1 pt) ⎡ Library/Rochester/setLinearAlgebra3Matrices/ur ⎤la 3 11.pg-1 -3 2Let M = ⎣ -3 0 -2 ⎦ .3 1 -3Find c 1 , c 2 , and c 3 such that M 3 +c 1 M 2 +c 2 M +c 3 I 3 = 0, whereI 3 is the identity 3 × 3 matrix.c 1 = ,c 2 = ,c 3 = .Answer(s) submitted:•••(incorrect)8. (1 pt) Library/Rochester/setLinearAlgebra22SymmetricMatrices-/ur la 22 6.pg ⎡⎤4 -2 0 0The matrix M = ⎢ -2 4 0 0⎥⎣ 0 0 4 -2 ⎦ .0 0 -2 4has two distinct eigenvalues λ 1 < λ 2 . Find the eigenvalues andan orthonormal basis for each eigenspace.λ 1 = ,⎡ ⎤ ⎤associated unit eigenvector = ⎢⎣λ 2 = ,⎡⎥⎦ , ⎢⎣⎥⎦ ,⎡associated unit eigenvector = ⎢⎣⎤ ⎡⎥⎦ , ⎢⎣⎤⎥⎦ .The above eigenvectors form an orthonormal eigenbasis for M.Answer(s) submitted:••••(incorrect)9. (1 pt) Library/TCNJ/TCNJ Eigenvalues/problem2.pgA is an n × n matrix.Check the true statements below:• A. A steady-state vector for a stochastic matrix is actuallyan eigenvector.• B. The eigenvalues of a matrix are on its main diagonal.• C. An eigenspace of A is just a null space of a certainmatrix.• D. If Ax = λx for some vector x, then x is an eigenvectorof A.• E. If v 1 and v 2 are linearly independent eigenvectors,then they correspond to distinct eigenvalues.Answer(s) submitted:•(incorrect)Generated by c○WeBWorK, http://webwork.maa.org, Mathematical Association of America2