{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Math 54 - Quiz 09 Ans, Spring 2006

Math 54 - Quiz 09 Ans, Spring 2006 - Therefore the general...

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Math 54, Quiz 9 Solutions Sections 202/203 GSI Carter 1. Write the word “true” or “false”: Let A be a 3 × 3 matrix with real entries and complex eigenvalue α + βi (that is, α is a real number and β is a nonzero real number). Then A has three distinct eigenvalues. True. The roots of the characteristic polynomial of A must be α + βi , α - βi , and some real number. 2. (a) Solve the following initial value problem: x = 2 4 4 2 x x (0) = 0 1 The characteristic polynomial of the matrix is ( λ - 2) 2 - 16 = λ 2 - 4 λ - 12 = ( λ + 2)( λ - 6) , so the eigenvalues are - 2 and 6. The eigenspace for - 2 is the null space of - 4 - 4 - 4 - 4 which is spanned by (1 , - 1) T . The eigenspace for 6 is the null space of 4 - 4
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: . Therefore the general solution to the system of differential equations is given by y = c 1 ± 1-1 ² e-2 t + c 2 ± 1 1 ² e 6 t . From the initial condition, we have that ± 1 ² = c 1 ± 1-1 ² + c 2 ± 1 1 ² , and therefore c 1 =-1 2 and c 2 = 1 2 . Therefore the solution to the initial value problem is given by y =-1 2 ± 1-1 ² e-2 t + 1 2 ± 1 1 ² e 6 t . (b) What type of system is the above differential equation? Circle one. i. source ii. sink iii. saddle point iv. spiral in v. spiral out vi. periodic Since the eigenvalues are real and have opposite signs, it is a saddle point. 1...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern