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**Roam** There is an invertible matrix P for which $\displaystyle P^{-1}AP$ is a diagonal matrix.

This may be far from true: not every matrix is diagonalizable.

Tonio

If P exists then A is diagonalizable. Do you know how to find P? Have you learnt about diagonalizability? Your 3x3 matrix A is diagonalizable if A has 3 linearly independent eigenvectors. So, the first step you have to find eigenvectors of A, say $\displaystyle P_1, P_2, P_3$. Then form the matrix $\displaystyle P=[P_1,P_3,P_3]$. the matrix $\displaystyle P^{-1}AP$ will be diagonal and upper triangular and will have the eigenvalues corresponding to $\displaystyle P_1, P_2, P_3$, respectively, as its successive diagonal entries. Let's see how you go.