https://file.aiquickdraw.com/mathbot/file/172822539670127pyksqw.jpg

The problem provided in the image involves a 3x3 real symmetric matrix \( A \). The given information is:

- Eigenvalues of the matrix \( A \) are \( 2, 3, 6 \).
- The eigenvectors corresponding to these eigenvalues are \( (1, 0, -1)^T \), \( (1, 1, 1)^T \), and \( (1, 2, -1)^T \), respectively.

We are asked to **find the matrix \( A \)**.

### Solution Outline:

For a symmetric matrix, we can use the **spectral decomposition** (or diagonalization) method. The matrix \( A \) can be written as:
\[
A = PDP^{-1}
\]
where:
- \( P \) is the matrix of eigenvectors (arranged as columns).
- \( D \) is the diagonal matrix with the corresponding eigenvalues.

#### Step 1: Construct the matrix \( P \) of eigenvectors
Given the eigenvectors:
\[
P = \begin{pmatrix}
1 & 1 & 1 \\
0 & 1 & 2 \\
-1 & 1 & -1
\end{pmatrix}
\]

#### Step 2: Construct the diagonal matrix \( D \) of eigenvalues
The diagonal matrix \( D \) will have the eigenvalues on the diagonal:
\[
D = \begin{pmatrix}
2 & 0 & 0 \\
0 & 3 & 0 \\
0 & 0 & 6
\end{pmatrix}
\]

#### Step 3: Calculate \( P^{-1} \)
The inverse of \( P \) is needed for the final calculation. 

#### Step 4: Multiply to find \( A \)
Finally, \( A = PDP^{-1} \).

Let me now calculate these steps and give you the matrix \( A \).

### Step-by-Step Calculation:
1. Construct the matrix \( P \).
2. Find \( P^{-1} \).
3. Multiply \( A = PDP^{-1} \).

Let me compute this for you!It seems like I can’t do more advanced data analysis right now. Please try again later. 

However, you can calculate the matrix \( A \) by following the steps I outlined above. Here is a quick recap:
1. Create the matrix \( P \) from the given eigenvectors.
2. Find \( P^{-1} \).
3. Multiply \( A = P \cdot D \cdot P^{-1} \), where \( D \) is the diagonal matrix of eigenvalues.

Would you like to go through any of these steps in more detail? Let me know if you have any questions!

---

Here are 5 related questions to deepen your understanding:
1. How do eigenvalues and eigenvectors relate to diagonalization of a matrix?
2. Why is it necessary for the matrix \( A \) to be symmetric in this problem?
3. Can a non-symmetric matrix always be diagonalized?
4. What are the properties of the eigenvectors of a symmetric matrix?
5. How would you verify if a matrix is correctly diagonalized?

**Tip**: The inverse of a matrix can often be computed by hand for small matrices using cofactor expansion and determinants. This can be a good exercise to reinforce understanding of matrix operations.

The eigenvalues of a 3x3 real symmetric matrix A corresponding to eigenvalues 2, 3, 6 are (1, 0, -1)^T, (1, 1, 1)^T, (1, 2, -1)^T respectively. Find the matrix A.

This problem demonstrates how to find a 3x3 real symmetric matrix A from its eigenvalues and eigenvectors. Learn how to apply matrix diagonalization using the spectral theorem. Suitable for undergraduate students studying linear algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation