Math Problem Statement

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

Solution

To find the eigenvalues of the matrix B=(6223)B = \begin{pmatrix} 6 & 2 \\ 2 & 3 \end{pmatrix}, we need to solve the characteristic equation, which is given by:

det(BλI)=0\det(B - \lambda I) = 0

Where λ\lambda is an eigenvalue, and II is the identity matrix. Let's solve this step by step.

  1. Matrix BλIB - \lambda I: BλI=(6223)λ(1001)=(6λ223λ)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(BλI)=det((6λ223λ))=(6λ)(3λ)(2)(2)\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: (6λ)(3λ)4=186λ3λ+λ24=λ29λ+14=0(6 - \lambda)(3 - \lambda) - 4 = 18 - 6\lambda - 3\lambda + \lambda^2 - 4 = \lambda^2 - 9\lambda + 14 = 0

  3. Solving the quadratic equation: λ29λ+14=0\lambda^2 - 9\lambda + 14 = 0 We can solve this quadratic equation using the quadratic formula: λ=b±b24ac2a\lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} where a=1a = 1, b=9b = -9, and c=14c = 14.

    Substituting these values: λ=9±(9)24(1)(14)2(1)=9±81562=9±252=9±52\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: λ1=9+52=7andλ2=952=2\lambda_1 = \frac{9 + 5}{2} = 7 \quad \text{and} \quad \lambda_2 = \frac{9 - 5}{2} = 2

The eigenvalues of the matrix BB are λ1=7\lambda_1 = 7 and λ2=2\lambda_2 = 2.

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


Related Questions:

  1. How do you find the eigenvectors corresponding to the eigenvalues λ1=7\lambda_1 = 7 and λ2=2\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.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Eigenvalues
Determinants

Formulas

Characteristic Equation: det(B - λI) = 0
Quadratic Formula: λ = (-b ± √(b² - 4ac)) / 2a

Theorems

Eigenvalue Theorem

Suitable Grade Level

Undergraduate