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The image you uploaded provides a tutorial on applying eigenvalues and eigenvectors using the GeoGebra software. It visually demonstrates how the matrix transformation \(T(\vec{v}) = A \vec{v}\), where \(A\) is a matrix and \(\vec{v}\) is a vector, preserves the direction of vectors (eigenvectors) but scales them by their eigenvalues. Let’s break it down:

### Key Points from the Image:

1. **Matrix \(A\):**
   \[
   A = \begin{pmatrix} 1 & 2 \\ 4 & 3 \end{pmatrix}
   \]
   This matrix transforms vectors in a 2D plane.

2. **Eigenvalue Calculation:**
   The eigenvalues \(\lambda\) are solutions of the characteristic equation:
   \[
   \text{det}(A - \lambda I) = 0
   \]
   where \(I\) is the identity matrix. This leads to:
   \[
   \text{det}\begin{pmatrix} 1 - \lambda & 2 \\ 4 & 3 - \lambda \end{pmatrix} = 0
   \]
   Solving this equation gives the eigenvalues \( \lambda_1 = -1 \) and \( \lambda_2 = 6 \).

3. **Eigenvectors Calculation:**
   For each eigenvalue \(\lambda\), solve \( (A - \lambda I) \vec{v} = 0 \) to find the associated eigenvectors.

   - For \(\lambda_1 = -1\):
     \[
     (A - (-1)I)\vec{v} = 0
     \]
     This gives eigenvector \(\vec{v}_1 = \begin{pmatrix} -2 \\ 1 \end{pmatrix}\).

   - For \(\lambda_2 = 6\):
     \[
     (A - 6I)\vec{v} = 0
     \]
     This gives eigenvector \(\vec{v}_2 = \begin{pmatrix} 1 \\ 2 \end{pmatrix}\).

4. **Geometric Representation in GeoGebra:**
   The vectors are represented graphically, showing how the matrix \(A\) transforms the vectors. The red and blue arrows in the diagrams likely represent the eigenvectors associated with the eigenvalues.

5. **Interactive Step in GeoGebra:**
   The tutorial suggests manipulating a red slider to observe how eigenvalue calculations change dynamically as you adjust parameters, showing the eigenvalue problem's results on the right.

### Procedure to Apply:

- Substituting \(\lambda_1 = -1\) and \(\lambda_2 = 6\) back into the system gives the eigenvectors \(\vec{v}_1\) and \(\vec{v}_2\).
- Follow the step-by-step instructions in the GeoGebra software to explore eigenvector behavior visually.

Would you like a detailed breakdown of eigenvalue and eigenvector calculations from this matrix? Let me know if you have further questions.

Here are some follow-up questions you might find helpful:
1. How do eigenvectors help in understanding matrix transformations?
2. What are the typical applications of eigenvalues and eigenvectors in real-world scenarios?
3. How does GeoGebra assist in visualizing linear transformations?
4. Can you explore eigenvalues and eigenvectors of a 3x3 matrix using GeoGebra?
5. How can changes in a matrix's entries affect its eigenvalues and eigenvectors?

**Tip:** Eigenvectors represent directions in which a transformation acts by merely scaling the vector, while eigenvalues indicate the factor by which the vector is scaled.

Aplicação de autovalores e autovetores com o uso do software GeoGebra

Descubra como calcular autovalores e autovetores utilizando o software GeoGebra, com exemplos práticos e representações gráficas. Aprenda passo a passo como uma matriz transforma vetores no plano e explore a interação dinâmica com o software.

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