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To find the eigenvalues of the matrix \( B = \begin{pmatrix} 6 & 2 \\ 2 & 3 \end{pmatrix} \), we need to solve the characteristic equation, which is given by:

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

Where \( \lambda \) is an eigenvalue, and \( I \) is the identity matrix. Let's solve this step by step.

1. **Matrix \( B - \lambda I \):**
   \[
   B - \lambda I = \begin{pmatrix} 6 & 2 \\ 2 & 3 \end{pmatrix} - \lambda \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 6 - \lambda & 2 \\ 2 & 3 - \lambda \end{pmatrix}
   \]

2. **Characteristic equation:**
   \[
   \det(B - \lambda I) = \det\left(\begin{pmatrix} 6 - \lambda & 2 \\ 2 & 3 - \lambda \end{pmatrix}\right) = (6 - \lambda)(3 - \lambda) - (2)(2)
   \]
   Expanding this determinant:
   \[
   (6 - \lambda)(3 - \lambda) - 4 = 18 - 6\lambda - 3\lambda + \lambda^2 - 4 = \lambda^2 - 9\lambda + 14 = 0
   \]

3. **Solving the quadratic equation:**
   \[
   \lambda^2 - 9\lambda + 14 = 0
   \]
   We can solve this quadratic equation using the quadratic formula:
   \[
   \lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   where \( a = 1 \), \( b = -9 \), and \( c = 14 \).

   Substituting these values:
   \[
   \lambda = \frac{9 \pm \sqrt{(-9)^2 - 4(1)(14)}}{2(1)} = \frac{9 \pm \sqrt{81 - 56}}{2} = \frac{9 \pm \sqrt{25}}{2} = \frac{9 \pm 5}{2}
   \]
   So, the solutions are:
   \[
   \lambda_1 = \frac{9 + 5}{2} = 7 \quad \text{and} \quad \lambda_2 = \frac{9 - 5}{2} = 2
   \]

**The eigenvalues of the matrix \( B \) are \( \lambda_1 = 7 \) and \( \lambda_2 = 2 \).**

Would you like a detailed explanation of how to find the eigenvectors, or do you have any other questions?

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**Related Questions:**
1. How do you find the eigenvectors corresponding to the eigenvalues \( \lambda_1 = 7 \) and \( \lambda_2 = 2 \)?
2. What is the geometric interpretation of eigenvalues and eigenvectors?
3. Can the eigenvalues be negative, and what would that signify?
4. How do eigenvalues relate to the determinant and trace of a matrix?
5. What are the implications if a matrix has repeated eigenvalues?

**Tip:** Eigenvalues provide crucial insights into the properties of a matrix, especially in applications involving stability and oscillation in systems of differential equations.

For the matrix B = [ [6, 2], [2, 3] ], what are the eigenvalues?

This page provides a step-by-step solution to find the eigenvalues of the 2x2 matrix B = [ [6, 2], [2, 3] ]. Learn about key concepts in linear algebra, including how to solve the characteristic equation and apply the quadratic formula. Suitable for undergraduate students studying linear algebra.

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