Exercise · ZW-LINEAR-SYSTEM-AND-EIGENVALUES-008 · ★★☆ Intermediate
If \(A=\begin{pmatrix}0&1\\-4&0\end{pmatrix}\), its eigenvalues are \(\pm2i\). How do the solutions behave?
Choose one option.
Show answer
- B. Oscilan con período \(\pi\)
Explanation
With \(\lambda=\pm2i\), \(\cos2t\) and \(\sin2t\) appear, with no real part to damp or amplify them. The period is \(2\pi/2=\pi\).
Review the concept: Linear system and eigenvalues
Homogeneous system with a constant \(n\times n\) matrix \(A\), whose elementary solutions are built from an eigenvalue \(\lambda\) and an eigenvector \(\mathbf v\) of \(A\).
\[\mathbf x'=A\mathbf x,\qquad \mathbf x(t)=e^{\lambda t}\mathbf v,\quad A\mathbf v=\lambda\mathbf v\]
More practice problems: linear system and eigenvalues
- ZW-LINEAR-SYSTEM-AND-EIGENVALUES-001 What are the eigenvalues of A=[1 2; 2 1]?
- ZW-LINEAR-SYSTEM-AND-EIGENVALUES-002 For A=[1 2; 2 1], which eigenvector corresponds to λ=3?
- ZW-LINEAR-SYSTEM-AND-EIGENVALUES-003 The general solution of mathbf x'=[1 2; 2 1]mathbf x is:
- ZW-LINEAR-SYSTEM-AND-EIGENVALUES-004 Solve mathbf x'=[2 0; 0 -5]mathbf x with mathbf x(0)=(1,1).
- ZW-LINEAR-SYSTEM-AND-EIGENVALUES-005 For A=[2 1; 1 2], which is a solution of mathbf x'=Amathbf x?
- ZW-LINEAR-SYSTEM-AND-EIGENVALUES-006 If A=[-1 0; 0 -3], what happens to the solutions of mathbf x'=Amathbf x as t→∞?
- ZW-LINEAR-SYSTEM-AND-EIGENVALUES-007 For A=[0 1; -2 -3], the solution of mathbf x'=Amathbf x with mathbf x(0)=(1,0) has first component:
- ZW-LINEAR-SYSTEM-AND-EIGENVALUES-009 A 2×2 matrix has eigenvalues 2 and -3. What are its trace and determinant?
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