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We are given the matrix \( A = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \) and we are asked to find the determinant of \( B = A^{11} \), i.e., \( \det(B) \).

### Steps:
1. **Properties of the determinant**:
   - The determinant of a product of matrices equals the product of the determinants: \( \det(A^n) = (\det(A))^n \).
   
2. **Determinant of \( A \)**:
   - The matrix \( A = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \) is a diagonal matrix, and the determinant of a diagonal matrix is the product of the diagonal elements. Hence,
   \[
   \det(A) = 1 \times (-1) = -1.
   \]

3. **Determinant of \( B \)**:
   - Since \( B = A^{11} \), we can apply the property \( \det(B) = \det(A^{11}) = (\det(A))^{11} \).
   \[
   \det(B) = (-1)^{11} = -1.
   \]

Thus, the determinant of \( B \) is \( \boxed{-1} \).

Would you like further details or clarifications?

### Related Questions:
1. What is the determinant of \( A^{12} \)?
2. How do you compute the determinant of a general diagonal matrix?
3. What is the geometric interpretation of the determinant of a matrix?
4. How would the answer change if \( B = A^n \) for a general \( n \)?
5. Can you prove the determinant property \( \det(A^n) = (\det(A))^n \)?

### Tip:
For diagonal matrices, the determinant is always the product of the diagonal elements, making the computation very straightforward!

Let A = [[1, 0], [0, -1]]. Let B = A^11. Find the determinant of B.

Learn how to compute the determinant of B = A^11, where A is a diagonal matrix [[1, 0], [0, -1]]. This problem applies the properties of matrix algebra and determinants to arrive at the solution, which is -1. Suitable for students in Grades 10-12.

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