Question

A is the set of all matrices of order 3 with entries 0 or 1 only. Let \( B \subset A \) be the subset consisting of all matrices with determinant value 1. Let \( C \subset A \) be the subset consisting of all matrices with determinant value -1. Then, which of the following is true?

• A. \( A = B \cup C \)
• B. \( C \) is empty
• C. \( B \) and \( C \) contain the same number of elements
• D. \( B \) has twice as many elements as \( C \)

Hint:

Remember: Flipping the sign of a row in a matrix multiplies the determinant by -1. Use this symmetry when comparing matrices with determinant 1 and -1.

Answer 1

The Correct Option is C

Step 1: Each entry in a \( 3 \times 3 \) matrix from set \( A \) is either 0 or 1. So the total number of such matrices is: \[ |A| = 2^{3 \times 3} = 2^9 = 512 \] Step 2: Among these matrices: - Some matrices will have determinant 0 (non-invertible), - Some will have determinant 1 (subset \( B \)), - Some will have determinant -1 (subset \( C \)). 
Step 3: For each matrix with determinant 1, multiplying any one row by -1 flips the sign of the determinant. Thus, for each matrix in \( B \), there exists a corresponding matrix in \( C \), and vice versa. 
Step 4: Therefore, \( |B| = |C| \).