The problem in the image involves finding the eigenvalues and eigenvectors of a 2x2 matrix $A$. I'll walk you through the steps based on the image provided.

Step 1: Matrix $A$

The matrix $A$ given is: $A = \begin{pmatrix} 2 & 2 \\ 1 & 3 \end{pmatrix}$

Step 2: Finding the Eigenvalues

To find the eigenvalues $\lambda$, we solve the characteristic equation: $\det(A - \lambda I) = 0$ Here, $I$ is the identity matrix: $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ So, $A - \lambda I$ becomes: $A - \lambda I = \begin{pmatrix} 2 - \lambda & 2 \\ 1 & 3 - \lambda \end{pmatrix}$ The determinant of this matrix is: $\det(A - \lambda I) = (2 - \lambda)(3 - \lambda) - (2)(1)$ Expanding the determinant: $\det(A - \lambda I) = (2 - \lambda)(3 - \lambda) - 2 = (6 - 5\lambda + \lambda^2) - 2 = \lambda^2 - 5\lambda + 4$ Setting the determinant to zero to find the eigenvalues: $\lambda^2 - 5\lambda + 4 = 0$ This quadratic equation can be factored as: $(\lambda - 1)(\lambda - 4) = 0$ So, the eigenvalues are: $\lambda_1 = 1, \quad \lambda_2 = 4$

Step 3: Finding the Eigenvectors

Eigenvector for $\lambda_1 = 1$:

Substitute $\lambda = 1$ into $(A - \lambda I)\mathbf{v} = 0$: $(A - I)\mathbf{v} = \begin{pmatrix} 1 & 2 \\ 1 & 2 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = 0$ This simplifies to the system of equations: $x_1 + 2x_2 = 0$ The solution is: $\mathbf{v}_1 = \begin{pmatrix} -2 \\ 1 \end{pmatrix}$

Eigenvector for $\lambda_2 = 4$:

Substitute $\lambda = 4$ into $(A - \lambda I)\mathbf{v} = 0$: $(A - 4I)\mathbf{v} = \begin{pmatrix} -2 & 2 \\ 1 & -1 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = 0$ This simplifies to the system of equations: $-2x_1 + 2x_2 = 0$ The solution is: $\mathbf{v}_2 = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$

Final Answer

• The eigenvalues are $\lambda_1 = 1$ and $\lambda_2 = 4$.
• The corresponding eigenvectors are:
  • For $\lambda_1 = 1$, the eigenvector is $\mathbf{v}_1 = \begin{pmatrix} -2 \\ 1 \end{pmatrix}$.
  • For $\lambda_2 = 4$, the eigenvector is $\mathbf{v}_2 = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$.

Would you like more details, or do you have any questions?

Related Questions:

1. How do you verify if a vector is an eigenvector of a matrix?
2. What is the geometric interpretation of eigenvalues and eigenvectors?
3. Can a matrix have complex eigenvalues and eigenvectors? If yes, when?
4. What does it mean if a matrix has repeated eigenvalues?
5. How are eigenvalues used in diagonalization of matrices?
6. What are some applications of eigenvalues and eigenvectors in real-world problems?
7. How do you find eigenvalues and eigenvectors for larger matrices (e.g., 3x3)?
8. Can a non-square matrix have eigenvalues and eigenvectors?

Tip:

Eigenvectors corresponding to different eigenvalues are linearly independent, which is useful in matrix diagonalization.