Question

Find the characteristic polynomial of the 3x3 matrix.

Answer 1

Given a 3x3 matrix:
\[A = \left[\begin{matrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{matrix}\right]\]
Simply take:
\[\det(A-\lambda I) = \det\left[\begin{matrix}a_{11} - \lambda & a_{12} & a_{13} \\ a_{21} & a_{22}-\lambda & a_{23} \\ a_{31} & a_{32} & a_{33}-\lambda\end{matrix}\right]\]
This will give you a polynomial in lambda. Solve for that and you have your:
\[\boxed{\lambda_{1,2,3} \equiv\text{ eigenvalues}}\]

Answer 2

Setting that determinant to zero that is.

Answer 3

Oh thanks. How do I find the minimal polynomial?