Exercise · ZW-LINEAR-SYSTEM-AND-EIGENVALUES-003 · ★☆☆ Basic
The general solution of \(\mathbf x'=\begin{pmatrix}1&2\\2&1\end{pmatrix}\mathbf x\) is:
Choose one option.
Show answer
- D. \(c_1e^{3t}(1,1)+c_2e^{-t}(1,-1)\)
Explanation
Each (eigenvalue, eigenvector) pair contributes \(e^{\lambda t}\mathbf v\): \(\lambda=3\) with \((1,1)\) and \(\lambda=-1\) with \((1,-1)\). The vectors must not be swapped.
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-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-008 If A=[0 1; -4 0], its eigenvalues are pm2i. How do the solutions behave?
- 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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