Consider the set A of all matrices of order 3 `xx` 3 with entries 0 or 1 only. Let B be the subset of A consisting of all matrices whose determinant is 1. Let C be the subset of A consisting of all matrices whose determinant is -1. Then which one of the following is correct ?

Options

A. C is empty
B. B has as many elements as C
C. A=B `uu` C
D. B has twice as many elements as C

Answer

To solve the problem, we need to analyze the sets A, B, and C as defined in the question.

### Step-by-Step Solution:

1. **Understanding Set A**:
- Set A consists of all 3x3 matrices with entries either 0 or 1.
- The total number of such matrices can be calculated as follows:
- Each entry in a 3x3 matrix has 2 choices (0 or 1).
- Therefore, the total number of matrices in A is \(2^{3 \times 3} = 2^9 = 512\).

2. **Understanding Set B**:
- Set B is a subset of A consisting of all matrices whose determinant is 1.
- For a 3x3 matrix to have a determinant of 1, it must be invertible and have full rank (rank 3).
- The matrices that can achieve this must have a specific arrangement of 1's and 0's that allows for a non-zero determinant.

3. **Understanding Set C**:
- Set C is a subset of A consisting of all matrices whose determinant is -1.
- Similar to set B, for a matrix to have a determinant of -1, it must also be invertible.
- However, the determinant of a matrix can change sign based on row or column operations (specifically, interchanging rows or columns).

4. **Relationship Between Sets B and C**:
- If a matrix in set B has a determinant of 1, then by interchanging two rows (or columns), we can create a matrix in set C with a determinant of -1.
- Thus, for every matrix in B, there is a corresponding matrix in C obtained by interchanging two rows or columns.

5. **Conclusion**:
- Since interchanging rows or columns changes the sign of the determinant, the number of matrices in set B (determinant = 1) is equal to the number of matrices in set C (determinant = -1).
- Therefore, the correct statement is that the number of elements in B is equal to the number of elements in C.

### Final Answer:
The correct option is that the number of elements in set B is equal to the number of elements in set C.