Systems of Differential Equations and Eigenvalue Solutions · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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One matrix, two linked unknowns

Mathematics · Differential Equations · ages 21-22
Name ______________________   Date ____________

  1. A diagonal matrix has 4 and 7 on its diagonal. What is its larger eigenvalue?

    Answer: ______________

  2. What are the eigenvalues of the matrix with rows 4 1 and 2 3?

    • 2 and 5
    • 4 and 3
    • -2 and -5
    • 7 and 10

  3. Convert y double prime plus 5 y prime plus 6y equals 0 into a first order system with v equals y prime. What is it?

    • y prime equals v, v prime equals minus 6y minus 5v
    • y prime equals v, v prime equals 6y plus 5v
    • y prime equals v, v prime equals minus 5y minus 6v
    • y prime equals v, v prime equals y plus v

  4. What are the eigenvalues of the diagonal two three matrix?

    • 1 and 6
    • 2 and 3
    • 0 and 5

  5. The vector one one is an eigenvector of the diagonal two three matrix.

    Circle one:   True   False

  6. How does y double prime plus four y prime plus three y become a system?

    • Set v as y prime, then v prime is three y plus four v
    • Set v as y prime, then v prime is minus three y minus four v
    • Set v as y double prime, then v prime is minus three y minus four v

  7. Why is an axis direction an eigenvector of a diagonal matrix?

    • The matrix only stretches it, never turns it
    • Its entries off the diagonal are the eigenvalues
    • Every vector is an eigenvector of a diagonal matrix

  8. What equation gives the growth rates?

    • Trace plus one equals zero
    • Determinant of A minus lambda I equals zero
    • A times x equals b

  9. Type the sum of the eigenvalues of the diagonal two three matrix.

    Answer: ______________

  10. A claim says three one is an eigenvector. What is the honest check?

    • Check whether entries look nice
    • Check that the eigenvalue is a whole number
    • Multiply and check parallel
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Answer key

For grown-ups. Fold this page away before handing over the rest.

One matrix, two linked unknowns W1-mt_BbOqJ4S1ur-s1

  1. 7 · The eigenvalues are the diagonal entries, and 7 exceeds 4.
  2. 2 and 5 · (4 - lambda)(3 - lambda) - 2 = lambda squared - 7 lambda + 10 = 0, so lambda is 2 or 5.
  3. y prime equals v, v prime equals minus 6y minus 5v · Solve the original for the highest derivative: v prime equals minus 6y minus 5v.
  4. 2 and 3 · Diagonal entries are the eigenvalues, sitting in plain sight.
  5. False · Multiplying returns two three, which points another way.
  6. Set v as y prime, then v prime is minus three y minus four v · Velocity carries the first order, and the old equation gives the second line.
  7. The matrix only stretches it, never turns it · Stretching without turning is what defines an eigenvector.
  8. Determinant of A minus lambda I equals zero · The characteristic equation comes from zeroing that determinant.
  9. 5 · Two plus three is five.
  10. Multiply and check parallel · Only the multiply test decides, since parallel is the whole definition.
Worksheet · LightMySky