Exercise · ZW-EIGENVALUE-AND-EIGENVECTOR-010 · ★★★ Advanced
A \(3\times3\) matrix has eigenvectors \(v_1,v_2,v_3\) with eigenvalues 2, 3 and 5. What is \(A(v_1+v_2)\)?
Choose one option.
Show answer
- A. \(2v_1+3v_2\)
Explanation
By linearity, \(A(v_1+v_2)=Av_1+Av_2=2v_1+3v_2\). The sum is not an eigenvector because the eigenvalues differ.
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?
- ZW-EIGENVALUE-AND-EIGENVECTOR-002 Let A=[1 2; 2 1] and v=(1,1). What is the eigenvalue associated with v?
- ZW-EIGENVALUE-AND-EIGENVECTOR-003 The vectors (1,0) and (1,-1) are eigenvectors of A=[3 1; 0 2]. What are their eigenvalues, in that…
- 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)?