Exercise · ZW-EIGENVALUE-AND-EIGENVECTOR-009 · ★★★ Advanced
If 0 is an eigenvalue of the square matrix \(A\), what follows?
Choose one option.
Show answer
- C. \(\det A=0\)
Explanation
If \(Av=0\cdot v=0\) with \(v\neq0\), the system \(Ax=0\) has nonzero solutions and hence \(\det A=0\). The other options do not follow: for example, \(\begin{pmatrix}1&1\\0&0\end{pmatrix}\) has eigenvalue 0 and is neither zero nor symmetric.
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)?
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