Practice Extended Example: 3×3 Matrix (32.10) - Basis of Eigenvectors

Practise this section

10 questions

Flashcard drill
4 cards from this section. Quick, good the night before a test.
Exercise
6 questions · about 4 min · pass mark 70%. Hints available.
Sign up to start the exercise
Quiz
4 questions · about 3 min · pass mark 70%. Hints available.
Sign up to start the quiz

6 sample questions

Question 1.
What is an eigenvector?
Show hintHide hint
Think about transformations applied to vectors.
Question 2.
What does the characteristic polynomial help us find?
Show hintHide hint
Recall the purpose of deriving this polynomial.
3Question 3.
What is the first step in finding the eigenvalues of a matrix?
- Calculate the trace of the matrix.
- Find the determinant of (A - λI).
- Directly calculate eigenvectors.
Show hintHide hint
Recall the importance of the determinant in linear transformations.
4Question 4.
True or False: A matrix is diagonalizable if it has n distinct eigenvalues.
- True
- False
Show hintHide hint
Consider matrix properties on multiplication and transformations.
Question 5.
Given the matrix A = \begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix}, find its eigenvalues and eigenvectors.
Show hintHide hint
Remember to calculate the determinant first.
Question 6.
Discuss the geometric interpretation of eigenvalues and eigenvectors in a physical system like pendulums.
Show hintHide hint
Relate eigenvalues to physical properties such as stiffness or mass in vibrations.

6 more questions in this section. Enroll free to practise them all.