Question

Find the general solution of x'=(1, 1, 2, 1, 2, 1, 2, 1, 1)x. (this is 3x3 matrix, 1, 1, 2 on the left, 1, 2, 1 in the middle, 2, 1, 1 on the right.)

Answer 1

\[x'=\begin{pmatrix}1&1&2\\1&2&1\\2&1&1\end{pmatrix}x~~?\]
Oh and by the way, it's easier to type out row by row in latex, so would you mind describing the matrix row at a time?

Answer 2

This is right.

Answer 3

Any particular problems with working it out? Like with the eigenvalue/vector-finding?

Answer 4

Do you know how to set up the next step?

Answer 5

Well the next step would be the find the determinant \(|A-\lambda I|\), and solving for the eigenvalues. Unless you're given them? save some time

Answer 6

I don't know how it will looks like. After that, I'll solve the problem on my own.

Answer 7

\[\begin{vmatrix}1-\lambda&1&2\\1&2-\lambda&1\\2&1&1-\lambda\end{vmatrix}=(1-\lambda)\begin{vmatrix}2-\lambda&1\\1&1-\lambda\end{vmatrix}-1\begin{vmatrix}1&2\\1&1-\lambda\end{vmatrix}+2\begin{vmatrix}1&2\\2-\lambda&1\end{vmatrix}\]
via a cofactor expansion along the first column.