To solve this problem, let's analyze the given information:

• Determinant of the matrix = -1
• Sum of eigenvalues = 0

For a 2x2 matrix, the sum and product of its eigenvalues relate directly to its trace and determinant:

1. Sum of eigenvalues is equal to the trace of the matrix (the sum of the diagonal elements). Here, the sum is given as 0.
2. Product of eigenvalues is equal to the determinant of the matrix. In this case, the determinant is given as -1.

Now, let the eigenvalues of the matrix be $\lambda_1$ and $\lambda_2$.

• From the given conditions:
  • $\lambda_1 + \lambda_2 = 0$ (sum of eigenvalues)
  • $\lambda_1 \times \lambda_2 = -1$ (determinant)

The only pair of numbers that satisfy both conditions are $\lambda_1 = 1$ and $\lambda_2 = -1$.

Rank of the matrix:

• Since the eigenvalues are non-zero (i.e., 1 and -1), the matrix is invertible, and its rank is 2.

Thus, the rank of the matrix is 2.

Would you like more details or have any questions?

Here are some related questions you might find useful:

1. What is the relationship between eigenvalues and the determinant of a matrix?
2. How do you compute the rank of a matrix using eigenvalues?
3. Can a matrix with a zero eigenvalue be full rank?
4. What is the geometric interpretation of eigenvalues?
5. How do the trace and determinant of a matrix relate to its eigenvalues?

Tip: Eigenvalues provide key insights into the structure of a matrix, including its rank, invertibility, and the type of transformations it represents.