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Cogito

Linear Algebra · Chapter 7 · Lesson 2

Practice Worksheet

Finding Eigenvalues

The characteristic equation, and where it comes from.

12 problems · about 25 minutes · N-VM.C.11, A-REI.B.4

NameDate

Figure — use these to answer the problems

The eigenvalues are printed below. Check them against the rule: they should add to the trace and multiply to the determinant.
ij
  • i-hat lands on(2, 1)
  • j-hat lands on(2, 3)
  • Determinant4
  • Eigenvalues4 and 1
a2
b2
c1
d3
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. Trace 12 and determinant 35. What is the larger eigenvalue?

    Answer

  2. Why must the determinant of A − λI be zero?

    • It has to crush a non-zero vector, and only a singular matrix does that.
    • It is a convention chosen to make the algebra tidy.

  3. Rows [3, 0] and [0, 8]. What is the trace?

    Answer

  4. Rows [3, 0] and [0, 8]. What is the larger eigenvalue?

    Answer

  5. Trace 10 and determinant 21. What is the larger eigenvalue?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. Eigenvalues 6 and 2. What is the determinant?

    Answer

  2. Eigenvalues 6 and 2. What is the trace?

    Answer

  3. Trace 0 and determinant 1. Are the eigenvalues real? 1 yes, 0 no.

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. Order the steps for finding eigenvalues and eigenvectors.

    Write 1 to 4 in the boxes to put these in order.

    • Write the characteristic equation from the trace and determinant
    • Solve it for the eigenvalues
    • Substitute each eigenvalue back into A minus lambda I
    • Solve that system for the eigenvector

  2. A triangular matrix with diagonal 2 and 9. What is the smaller eigenvalue?

    Answer

  3. The Trace

    Rows [6, 2] and [1, 5]. What is the trace?

    Answer

  4. The Missing One

    Trace 9 and one eigenvalue is 4. What is the other?

    Answer