Originally Posted by

**drangelpsyche** Hello,

I've been asked to find a 3x3 matrix A such that

1) $\displaystyle A^3 - 4A^2 + 5A - 2I = 0$

and

2) The eigenvectors of A do not span $\displaystyle R^3$

I believe the second condition implies that A is not diagonalizable, which means it does not contain a complete set of eigenvectors (I am only aware that nilpotent matrices fall under this category, are there others?) and I know that 1) can be expressed as:

$\displaystyle (A - I)^2(A-2I)$

but I cannot figure out how to work this out. Can anyone help?