Exercise · ZW-LINEAR-SYSTEM-AND-EIGENVALUES-010 · ★★★ Advanced
For \(A=\begin{pmatrix}1&1\\1&1\end{pmatrix}\), which is a nonzero constant solution of \(\mathbf x'=A\mathbf x\)?
Choose one option.
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- A. \((1,-1)\)
Explanation
A constant solution satisfies \(A\mathbf x=\mathbf 0\), that is, it is an eigenvector with \(\lambda=0\). Here \(A(1,-1)=(0,0)\), whereas \(A(1,1)=(2,2)\).
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 \(A\).
\[\mathbf x'=A\mathbf x,\qquad \mathbf x(t)=e^{\lambda t}\mathbf v,\quad A\mathbf v=\lambda\mathbf v\]
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