A find this eigenvalue eigenvalue b find a basis for

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(a) Find this eigenvalue. eigenvalue = , (b) Find a basis for the associated eigenspace. Answer: Note: To enter a basis into WeBWorK, place the entries of each vector inside of brackets, and enter a list of these vectors, separated by commas. (c) Find the Geometric Multiplicity (GM) of the eigenvalue GM = Solution: (a) p ( λ ) = - 4 - λ 0 0 0 - 4 - λ 0 - 2 0 - 4 - λ Expanding the determinant along the second column gives: p ( λ ) = ( - 4 - λ ) - 4 - λ 0 - 2 - 4 - λ = ( - 4 - λ )[( - 4 - λ )( - 4 - λ ) - 0 ] = ( - 4 - λ )( λ 2 + 8 λ + 16 ) = ( - 4 - λ )( λ + 4 ) 2 Thus λ = - 4 is an eigenvalue of Algebraic Multiplicity 3. (b) To find a basis for the associated eigenspace we need to find a basis of the nullspace of A - ( - 4 ) I . The RREF of A - ( - 4 ) I is 1 0 0 0 0 0 0 0 0 . Thus a basis for the nullspace consists of the vectors 0 1 0 , - 0 0 1 . (c) Since the dimension of the eigenspace is 2 (there are two vectors in the basis), we have that the Geometric Multiplicity of the eigenvalues is 2. Correct Answers: -4 0 -1 -1 , 0 0 1 2 6. (1 point) The matrix A = - 8 4 - 4 - 4 0 - 4 4 - 4 0 has two real eigenvalues, one of multiplicity 1 and one of multiplicity 2. Find the eigenvalues and a basis of each eigenspace. λ 1 = has multiplicity 1, with a basis of . λ 2 = has multiplicity 2, with a basis of . To enter a basis into WeBWorK, place the entries of each vector inside of brackets, and enter a list of these vectors, separated by commas. For instance, if your basis is 1 2 3 , 1 1 1 , then you would enter [1,2,3],[1,1,1] into the answer blank. Solution: The characteristic equation of the matrix A is λ ( λ + 4 ) 2 = 0. Thus the eigenvector λ 1 = 0 has Algebraic Multi- plicity 1 and the eigenvalue λ 2 = - 4 has Algebraic Multiplicity 2. The RREf of A - 0 I = A is 1 0 1 0 1 1 0 0 0 Thus a basis of the eigennspace associated to λ 1 = 0 is - 1 - 1 1 . The RREf of A + 4 I is 1 - 1 1 0 0 0 0 0 0 Thus a basis of the eigennspace associated to λ 2 = - 4 is - 1 0 1 , 1 1 0 . Correct Answers: 0 -1 -1 1 -4 -1 0 1 , 1 1 0 7. (1 point) Find a basis of the eigenspace associated with the eigenvalue 1 of the matrix A = 2 0 - 4 5 - 2 1 4 - 6 - 1 0 4 - 4 - 1 0 4 - 4 . Answer: To enter a basis into WeBWorK, place the entries of each vector inside of brackets, and enter a list of these vectors, separated by 2
commas. For instance, if your basis is 1 2 3 , 1 1 1 , then you would enter [1,2,3],[1,1,1] into the answer blank. Solution: The RREF of A - ( 1 ) I is 1 0 0 1 0 0 1 - 1 0 0 0 0 0 0 0 0 Thus x 2 and x 4 are free variables and a basis of the nullspace of A - ( 1 ) I is 0 1 0 0 , - 1 0 1 1 Correct Answers: -1 0 1 1 , 0 1 0 0 8. (1 point) Find the eigenvalues λ 1 < λ 2 and associated unit eigenvectors ~ u 1 ,~ u 2 of the symmetric matrix A = - 16 - 18 - 18 11 .

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