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The image asks to find the eigenvalues and eigenvectors of a matrix \( A \), where:

\[
A = \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix}
\]

### Step 1: Eigenvalue Calculation
To find the eigenvalues, we need to solve the characteristic equation:

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

where \( I \) is the identity matrix, and \( \lambda \) is the eigenvalue. For the given matrix \( A \):

\[
A - \lambda I = \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix} - \begin{pmatrix} \lambda & 0 \\ 0 & \lambda \end{pmatrix} = \begin{pmatrix} 1 - \lambda & -1 \\ 0 & 1 - \lambda \end{pmatrix}
\]

Now, calculate the determinant:

\[
\det(A - \lambda I) = (1 - \lambda)(1 - \lambda) - (0)(-1) = (1 - \lambda)^2 = 0
\]

Solving this equation:

\[
(1 - \lambda)^2 = 0 \quad \Rightarrow \quad \lambda = 1
\]

Thus, the matrix has a repeated eigenvalue of \( \lambda = 1 \).

### Step 2: Eigenvector Calculation
Next, we find the eigenvector corresponding to \( \lambda = 1 \). We solve:

\[
(A - \lambda I) \mathbf{v} = 0
\]

For \( \lambda = 1 \), the matrix \( A - I \) becomes:

\[
A - I = \begin{pmatrix} 1 - 1 & -1 \\ 0 & 1 - 1 \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 0 & 0 \end{pmatrix}
\]

Now solve for the eigenvector \( \mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} \):

\[
\begin{pmatrix} 0 & -1 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}
\]

This results in the system:

\[
-1 \cdot v_2 = 0 \quad \Rightarrow \quad v_2 = 0
\]

Thus, \( v_1 \) can be any non-zero value. Therefore, the eigenvector corresponding to \( \lambda = 1 \) is:

\[
\mathbf{v} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}
\]

### Summary:
- Eigenvalue: \( \lambda = 1 \)
- Eigenvector: \( \mathbf{v} = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \)

Would you like more details on these steps or any specific part?

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Here are 5 related questions to deepen your understanding:
1. How do we calculate the determinant of a 3x3 matrix for finding eigenvalues?
2. What happens when a matrix has complex eigenvalues?
3. Can a matrix have more than one linearly independent eigenvector for the same eigenvalue?
4. How are eigenvalues used in diagonalizing a matrix?
5. What is the physical significance of eigenvectors in real-world applications?

**Tip:** For a matrix with repeated eigenvalues, check if there are multiple linearly independent eigenvectors by examining the geometric multiplicity.

To find the eigenvalues and eigenvectors of matrix A = [[1, -1], [0, 1]].

This page explains how to calculate the eigenvalues and eigenvectors of the 2x2 matrix A = [[1, -1], [0, 1]]. It covers key concepts of linear algebra, including matrix operations and the characteristic equation. Suitable for undergraduate students.

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