give me A = 3 x 3 matrix question
finding a matrix P that orthogonally diagonalizes A

Here is a \( 3 \times 3 \) matrix problem:

### Problem:  
Find a matrix \( P \) that orthogonally diagonalizes the given matrix \( A \):

\[
A = \begin{bmatrix} 
4 & -2 & 2 \\ 
-2 & 5 & -4 \\ 
2 & -4 & 6 
\end{bmatrix}
\]

### Steps to Solve:  
1. Compute the **eigenvalues** of \( A \) by solving \( \det(A - \lambda I) = 0 \).  
2. Find the **orthonormal eigenvectors** corresponding to each eigenvalue.  
3. Construct \( P \) using the orthonormal eigenvectors as columns.  
4. Verify that \( P^T A P \) is a diagonal matrix \( D \).

Would you like me to solve this step by step?  

Here are 5 related questions:  
1. What does it mean for a matrix to be orthogonally diagonalizable?  
2. How do you check if a matrix is symmetric?  
3. What are the properties of an orthogonal matrix?  
4. Why do symmetric matrices always have orthogonal eigenvectors?  
5. What is the difference between diagonalization and orthogonal diagonalization?  

### **Tip:**  
A symmetric matrix (\( A^T = A \)) is always orthogonally diagonalizable!

give me A = 3 x 3 matrix question finding a matrix P that orthogonally diagonalizes A

Learn how to find a matrix that orthogonally diagonalizes a 3x3 matrix using eigenvalues and orthonormal eigenvectors. This solution explores key concepts in linear algebra, including diagonalization and orthogonal matrices, with a detailed, step-by-step approach.

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