Math 130 Final Quiz 4
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Math 130 Jekel
Final Quiz 4
March 22, 2023
Question 1 (3 points). Find the eigenvalues and eigenvectors of the matrix A = 
). What type
of equilibrium occurs at (0, 0) for the associated differential equation? If it is a spiral, say whether it is clockwise or counterclockwise rotation. Show your work.
Question 2 (3 points). Let A be a 2 × 2 matrix with eigenvalues and eigenvectors
λ 1 = −2, ⃗v1 = ( )2(1) , λ2 = −3, ⃗v3 = ( 
3)
Draw a phase portrait for the differential equation ⃗x\ (t) = A⃗x(t), including trajectories on the eigenlines and trajectories in each of the regions between the eigenlines.
Question 3 (2 points). Here is a plot of a vector field for a first order linear system in two dimensions. Several trajectories are shown in gray.
(a) What type of equilibrium does the system have at (0 , 0)?
(b) Is the determinant of the corresponding matrix positive, negative, or zero? Why?
Question 4 (2 points). Let A be a 2 × 2 real matrix and ⃗x(t) is a solution to ⃗x\ (t) = A⃗x(t). Suppose that ⃗x(t) stays in the first quadrant (positive x and y coordinates) for all t and that limt → ∞ ∥⃗x(t)∥ = ∞ .
(a) Is Tr(A)2 − 4det(A) ≥ 0? Yes / No / Insufficient information
(b) Is Tr(A) > 0? Yes / No / Insufficient information
Insufficient information means that either possibility could occur. For this question, showing your work is not required but will increase the probability of partial credit if you are wrong.
2023-03-25