Can you give me an example of how to calculate eigenvalues from a 3 by 3 matrix?

To calculate the eigenvalues of a 3x3 matrix, we follow a systematic procedure. Let’s go through an example:

### Step-by-Step Example:
Given the matrix \( A \):

\[
A = \begin{bmatrix}
4 & 1 & 1 \\
2 & 3 & 1 \\
1 & 1 & 2
\end{bmatrix}
\]

1. **Write the characteristic equation**:
   The eigenvalues of matrix \( A \) are the solutions to the characteristic equation:

   \[
   \det(A - \lambda I) = 0
   \]

   where \( I \) is the identity matrix, and \( \lambda \) represents the eigenvalue. The matrix \( A - \lambda I \) subtracts \( \lambda \) from the diagonal entries of \( A \):

   \[
   A - \lambda I = \begin{bmatrix}
   4-\lambda & 1 & 1 \\
   2 & 3-\lambda & 1 \\
   1 & 1 & 2-\lambda
   \end{bmatrix}
   \]

2. **Find the determinant of \( A - \lambda I \)**:
   The next step is to compute the determinant of \( A - \lambda I \):

   \[
   \det(A - \lambda I) = \det \begin{bmatrix}
   4-\lambda & 1 & 1 \\
   2 & 3-\lambda & 1 \\
   1 & 1 & 2-\lambda
   \end{bmatrix}
   \]

   Using cofactor expansion along the first row:

   \[
   \det(A - \lambda I) = (4-\lambda) \det \begin{bmatrix}
   3-\lambda & 1 \\
   1 & 2-\lambda
   \end{bmatrix}
   - 1 \cdot \det \begin{bmatrix}
   2 & 1 \\
   1 & 2-\lambda
   \end{bmatrix}
   + 1 \cdot \det \begin{bmatrix}
   2 & 3-\lambda \\
   1 & 1
   \end{bmatrix}
   \]

3. **Calculate the 2x2 determinants**:
   
   Let's calculate the 2x2 determinants:
   \[
   \det \begin{bmatrix}
   3-\lambda & 1 \\
   1 & 2-\lambda
   \end{bmatrix} = (3-\lambda)(2-\lambda) - 1 \cdot 1 = \lambda^2 - 5\lambda + 5
   \]
   
   \[
   \det \begin{bmatrix}
   2 & 1 \\
   1 & 2-\lambda
   \end{bmatrix} = 2(2-\lambda) - 1 \cdot 1 = 4 - 2\lambda - 1 = 3 - 2\lambda
   \]

   \[
   \det \begin{bmatrix}
   2 & 3-\lambda \\
   1 & 1
   \end{bmatrix} = 2(1) - (3-\lambda) \cdot 1 = 2 - 3 + \lambda = \lambda - 1
   \]

4. **Substitute these into the determinant expression**:
   Now we substitute back into the expansion:

   \[
   \det(A - \lambda I) = (4-\lambda)(\lambda^2 - 5\lambda + 5) - (3 - 2\lambda) + (\lambda - 1)
   \]

5. **Simplify the characteristic equation**:
   Expand and simplify:

   \[
   = (4-\lambda)(\lambda^2 - 5\lambda + 5) - 3 + 2\lambda + \lambda - 1
   \]
   
   \[
   = (4\lambda^2 - 20\lambda + 20 - \lambda^3 + 5\lambda^2 - 5\lambda) - 4 + 3\lambda
   \]
   
   \[
   = -\lambda^3 + 9\lambda^2 - 22\lambda + 16
   \]

6. **Solve for the eigenvalues**:
   The characteristic equation is:

   \[
   -\lambda^3 + 9\lambda^2 - 22\lambda + 16 = 0
   \]

   Solve this cubic equation to find the eigenvalues. By trial or using a solver, we find that the eigenvalues are:

   \[
   \lambda_1 = 4, \quad \lambda_2 = 2, \quad \lambda_3 = 1
   \]

### Final Eigenvalues:
The eigenvalues of the matrix \( A \) are:

\[
\lambda_1 = 4, \quad \lambda_2 = 2, \quad \lambda_3 = 1
\]

---

Would you like further details or clarification on any of the steps?

Here are 5 related questions to expand on this topic:

1. How can you calculate eigenvectors for the same matrix?
2. How are eigenvalues used in applications like physics or engineering?
3. How does the determinant of a matrix relate to its eigenvalues?
4. Can you explain the geometric interpretation of eigenvalues?
5. How do the properties of symmetric matrices affect their eigenvalues?

**Tip**: For larger matrices or complex eigenvalue problems, numerical methods like QR algorithm or power iteration are often used.

Learn how to calculate the eigenvalues of a 3x3 matrix with a detailed example. This guide explains how to write the characteristic equation, find the determinant, and solve for the eigenvalues. Suitable for students studying Linear Algebra at the undergraduate level.

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