Exercise · ZW-EIGENVALUE-AND-EIGENVECTOR-008 · ★★☆ Intermediate
If \(v\) is an eigenvector of \(A\) with eigenvalue 2, what is \(A(-3v)\)?
Choose one option.
Show answer
- B. \(-6v\)
Explanation
\(A(-3v)=-3\,Av=-3(2v)=-6v\). Also \(-3v\) is an eigenvector too, with the same eigenvalue 2.
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
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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-009 If 0 is an eigenvalue of the square matrix A, what follows?
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