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?

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.