Does the arrangement of eigenvectors matter for diagonalizing a matrix?

[pyroknife]

The problem is attached.
For these kind of problems where they find the matrix P that consists of the eigenvectors, does it matter the way the eigenvectors are arranged? Like can I interchange columns?

I know the determinant changes when I do that, but for diagonalizing purposes, is there a specific way the eigenvectors must be arranged in the matrix?

 

[HallsofIvy]

Yes, you can interchange colums (of course, P^-1 must be the multiplicative inverse of the new P). The result will be a diagonal matrix, still with the eigenvalues in different positions on the diagonal. Every eigenvector corresponds to a specific eigenvalue. What ever eigenvector you use as the first column will have its eigenvalue at the top left of the diagonal and so on.