Good! I hope you get a good grade! Since this particular matrix is "upper triangular", its eigenvalues are the numbers on the diagonal: 2, -1, and -1. The set of all eigenvectors for a given eigenvalue form a subspace (its "eigenspace"). The subspace for eigenvalue 2 is 1 dimensional but the eigenspace for eigenvalue might have dimension 2 or 1, depending on the number of independent eigenvectors. If there are two independent eigenvectors corresponding to eigenvalue -1 then the matrix is "diagonalizable" with the eigenvalues on the diagonal:
\begin{bmatrix}2 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}.
If all eigenvectors corresponding to an eigenvalue are multiples of one vector, so the "eigenspace" has dimension 1, the best we could do is the "Jordan Normal Form":
\begin{bmatrix}2 & 0 & 0 \\ 0 & -1 & 1 \\ 0 & 0 & -1 \end{bmatrix}.
It's easy to find a single eigenvector corresponding to eigenvalue 2: let \begin{bmatrix} x \\ y \\ z\end{bmatrix}. Then we must have \begin{bmatrix} 2 & -1 & 3 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix}= \begin{bmatrix}2x- y+ 3z \\ -y \\ -z\end{bmatrix}= 2\begin{bmatrix}x \\ y \\ z\end{bmatrix}= \begin{bmatrix}2x \\ 2y \\ 2z \end{bmatrix}. So we must have 2x- y+ 3z= 2x, -y= 2y, and -z= 2z. The last two equations give y= z= 0 so the first equation becomes 2x= 2x which is true for all x. Any eigenvector corresponding to eigenvalue 2 is of the form \begin{bmatrix} x \\ 0 \\ 0 \end{bmatrix}= x\begin{bmatrix}1 \\ 0 \\ 0 \end{bmatrix}.
So we look for eigenvectors corresponding to eigenvalue -1. If \begin{bmatrix}x \\ y \\ z \end{bmatrix} is such an eigenvector then we must have
\begin{bmatrix} 2 & -1 & 3 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix}= \begin{bmatrix}2x- y+ 3z \\ -y \\ -z\end{bmatrix}= -\begin{bmatrix} x \\ y \\ z\end{bmatrix}= \begin{bmatrix}-x \\ -y \\-z\end{bmatrix}.
So we must have 2x- y+ 3z= -x, -y= -y, and -z= -z. The last two equations are true for any y and z. 2x- y+ 3z= -x is the same as 3x- y+ 3z= 0 so y= 3x+ 3z. We can write any eigenvector as \begin{bmatrix} x \\ 3x+ 3z \\ z\end{bmatrix}= x\begin{bmatrix}1 \\ 3 \\ 0 \end{bmatrix}+ z\begin{bmatrix}0 \\ 3 \\ 1\end{bmatrix}. That is, the "eigenspace" corresponding to eigenvalue -1 has dimension 2 and is spanned by the two vectors \begin{bmatrix}1 \\ 3 \\ 0 \end{bmatrix} and \begin{bmatrix} 0 \\ 3 \\ 1 \end{bmatrix}. Since there are 3 independent eigenvectors, this matrix is diagonalizable.
Now, form the matrix, P, having those eigenvector as columns, P= \begin{bmatrix}1 & 1 & 0 \\0 & 3 & 3 \\ 0 & 0 & 1\end{bmatrix}. It's inverse matrix is P^{-1}= \begin{bmatrix}1 & -\frac{1}{3} & 1 \\ 0 & \frac{1}{3} & -1 \\ 0 & 0 & -1 \end{bmatrix}.
Then we have P^{-1}AP= \begin{bmatrix}1 & -\frac{1}{3} & 1 \\ 0 & \frac{1}{3} & -1 \\ 0 & 0 & -1 \end{bmatrix}\begin{bmatrix}2 & -1 & 3 \\ 0 & -1 & 0 \\ 0 & 0 & 1\end{bmatrix}\begin{bmatrix}1 & 1 & 0 \\0 & 3 & 3 \\ 0 & 0 & 1\end{bmatrix}
=\begin{bmatrix}2 & -\frac{2}{3} & 2 \\ 0 & -\frac{1}{3} & 1 \\ 0 & 0 & -1\end{bmatrix}\begin{bmatrix}1 & 1 & 0 \\0 & 3 & 3 \\ 0 & 0 & 1 \end{bmatrix}
= \begin{bmatrix}2 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}.