Exercise · ZW-EIGENVALUE-AND-EIGENVECTOR-004 · ★☆☆ Basic
Let \(A=\begin{pmatrix}2&1\\1&2\end{pmatrix}\). Which of these vectors is an eigenvector of \(A\)?
Choose one option.
Show answer
- B. \((1,-1)\)
Explanation
\(A(1,-1)=(2-1,\,1-2)=(1,-1)\), a multiple of itself (eigenvalue 1). The others give \((2,1)\), \((5,4)\) and \((4,5)\), which are not multiples of the original vector.
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-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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