6. Eigenvectors and Eigenvalues For the following matrix A: [5 A = 0 0 -3...

 

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6. Eigenvectors and Eigenvalues For the following matrix A: [5 A = 0 " 0 -3 4 0 3 i. Calculate A 0 and A 0 then comment on whether or not these vectors are eigenvectors 0 of A. If they are, give the corresponding eigenvalues. ii. Find the characteristic equation using the form D(A) = b3X³ + b₂λ² + b₁λ + bo and calculating the coefficients bo, b₁,b2, b3 using the formulas on page 17 of the study guide (Topic 6 and e.g., see online module 6.3. b₂ = -tr(A) in this case). iii. Now identify the characteristic equation D(X) in factorised form by first expanding the determinant XI – A. (Hint: you should be able to factor out one of the (A - aii) factors first, and then expand and factorise the resulting quadratic equation) - iv. Use the factorised form of D(A) to identify the eigenvalues of A, and then solve (AI-A)x=0 to find all eigenvectors not identified in part (i). 5 marks 6. Eigenvectors and Eigenvalues For the following matrix A: [5 A = 0 " 0 -3 4 0 3 i. Calculate A 0 and A 0 then comment on whether or not these vectors are eigenvectors 0 of A. If they are, give the corresponding eigenvalues. ii. Find the characteristic equation using the form D(A) = b3X³ + b₂λ² + b₁λ + bo and calculating the coefficients bo, b₁,b2, b3 using the formulas on page 17 of the study guide (Topic 6 and e.g., see online module 6.3. b₂ = -tr(A) in this case). iii. Now identify the characteristic equation D(X) in factorised form by first expanding the determinant XI – A. (Hint: you should be able to factor out one of the (A - aii) factors first, and then expand and factorise the resulting quadratic equation) - iv. Use the factorised form of D(A) to identify the eigenvalues of A, and then solve (AI-A)x=0 to find all eigenvectors not identified in part (i). 5 marks

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