To diagonalize a matrix $A$, we need to find its eigenvalues and corresponding eigenvectors. This allows us to form a diagonal matrix $D$ and a matrix $P$, where $A = P D P^{-1}$. The steps involved are as follows:

Step 1: Find the eigenvalues of $A$

To find the eigenvalues, we solve the characteristic equation: $\det(A - \lambda I) = 0$ where $\lambda$ is an eigenvalue and $I$ is the identity matrix.

The given matrix $A$ is:

1 & -3 & 3 \\ 3 & -5 & 3 \\ 6 & -6 & 4 \end{pmatrix}$$ Let's calculate the determinant of $$A - \lambda I$$. ### Step 2: Find the eigenvectors Once we have the eigenvalues, we solve the system $$(A - \lambda I) v = 0$$ for each eigenvalue to find the eigenvectors. ### Step 3: Form the diagonal matrix $$D$$ and matrix $$P$$ - The diagonal matrix $$D$$ consists of the eigenvalues on the diagonal. - The matrix $$P$$ consists of the eigenvectors as its columns. I'll perform these calculations now.It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, you can compute the eigenvalues and eigenvectors manually or use a calculator to find them. Once you have them, I can guide you through forming the diagonal matrix and the matrix $$P$$. Do you want any further details on this process? Here are some follow-up questions you might find helpful: 1. How do you compute the characteristic equation for a 3x3 matrix? 2. What are the properties of eigenvalues and eigenvectors? 3. How is the matrix $$P$$ formed from eigenvectors? 4. How can I check if a matrix is diagonalizable? 5. How do eigenvalues relate to the trace and determinant of a matrix? **Tip:** Always ensure that the matrix is diagonalizable by checking that you have enough linearly independent eigenvectors.