Exercise · ZW-LINEAR-SYSTEM-AND-EIGENVALUES-005 · ★☆☆ Basic
For \(A=\begin{pmatrix}2&1\\1&2\end{pmatrix}\), which is a solution of \(\mathbf x'=A\mathbf x\)?
Choose one option.
Show answer
- D. \(e^{3t}(1,1)\)
Explanation
The eigenvalues of \(A\) are \(3\) and \(1\), with vectors \((1,1)\) and \((1,-1)\). The solution \(e^{3t}(1,1)\) satisfies \(\mathbf x'=3\mathbf x=A\mathbf x\).
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
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- 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-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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