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The question asks to determine the rank of a \(2 \times 2\) matrix given that its eigenvalues are \(-1\) and \(1\), and its singular values are \(1\) and \(0\).

### Key Concepts:
1. **Eigenvalues**: These are the solutions to the characteristic equation \(\det(A - \lambda I) = 0\) where \(A\) is the matrix and \(\lambda\) is an eigenvalue.
2. **Singular Values**: These are the square roots of the eigenvalues of the matrix \(A^T A\), and they provide information about the matrix's rank and behavior under linear transformations.
3. **Rank of a Matrix**: The rank of a matrix is the number of non-zero singular values. It represents the number of linearly independent rows or columns.

### Given:
- Eigenvalues: \(-1\) and \(1\)
- Singular values: \(1\) and \(0\)

### Conclusion:
The singular values directly give information about the matrix's rank. Since one of the singular values is \(1\) and the other is \(0\), this means the matrix has **one non-zero singular value**. Hence, **the rank of the matrix is 1**.

### Correct answer: 
**Rank is 1**.

Would you like more details on how singular values and rank relate to eigenvalues?

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Here are five related questions to expand your understanding:
1. How are singular values calculated from a matrix?
2. Can a matrix have different eigenvalues and singular values?
3. How does the rank of a matrix affect its invertibility?
4. What is the relationship between a matrix's rank and its determinant?
5. How do eigenvalues reflect the geometric properties of a matrix?

**Tip:** The rank of a matrix tells you the dimension of the image of the transformation represented by the matrix. A rank-1 matrix flattens space into a line or plane.

Given, that the eigenvalues of a 2 x 2 matrix are -1,1 and its singular values are 1,0. What is the rank of the matrix?

This problem explores the rank of a 2x2 matrix with eigenvalues -1 and 1, and singular values 1 and 0. Learn how eigenvalues and singular values are related to the matrix rank and solve this linear algebra question. Suitable for undergraduate students.

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