Exercise · ZW-EIGENVALUE-AND-EIGENVECTOR-002 · ★☆☆ Basic
Let \(A=\begin{pmatrix}1&2\\2&1\end{pmatrix}\) and \(v=(1,1)\). What is the eigenvalue associated with \(v\)?
Choose one option.
Show answer
- A. \(3\)
Explanation
\(Av=(1+2,\,2+1)=(3,3)=3v\), so the eigenvalue is 3.
Review the concept: Eigenvalue and eigenvector
An eigenvector \(v\) of a square matrix \(A\) is a nonzero vector that \(A\) only stretches, shrinks or flips, without taking it off its line; the factor \(\lambda\) is its eigenvalue.
\[Av=\lambda v,\quad v\neq 0\]
More practice problems: eigenvalue and eigenvector
- ZW-EIGENVALUE-AND-EIGENVECTOR-001 Let A=[4 1; 2 3]. Which of these numbers is an eigenvalue of A?
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- ZW-EIGENVALUE-AND-EIGENVECTOR-004 Let A=[2 1; 1 2]. Which of these vectors is an eigenvector of A?
- ZW-EIGENVALUE-AND-EIGENVECTOR-005 Can the zero vector be an eigenvector?
- ZW-EIGENVALUE-AND-EIGENVECTOR-006 If Av=4v with v≠0, what is A^2v?
- ZW-EIGENVALUE-AND-EIGENVECTOR-007 How many real eigenvalues does the 90° rotation A=[0 -1; 1 0] have?
- ZW-EIGENVALUE-AND-EIGENVECTOR-008 If v is an eigenvector of A with eigenvalue 2, what is A(-3v)?
- ZW-EIGENVALUE-AND-EIGENVECTOR-009 If 0 is an eigenvalue of the square matrix A, what follows?
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