Question

Let \( \mathcal{M} \) be the set of all invertible \( 5 \times 5 \) matrices with entries 0 and 1. For each \( M \in \mathcal{M} \), let \( n_1(M) \) and \( n_0(M) \) denote the number of 1's and 0's in \( M \), respectively. Then \[ \min_{M \in \mathcal{M}} |n_1(M) - n_0(M)| = \]

• A. 1
• B. 3
• C. 5
• D. 15

Hint:

For combinatorial problems involving matrices, ensure that the matrix satisfies the invertibility condition, and use trial and error or optimization techniques for finding the minimum.

Answer 1

The Correct Option is A

Step 1: Understanding the problem.
We are asked to minimize the difference between the number of 1's and 0's in an invertible \( 5 \times 5 \) matrix with entries 0 and 1. An invertible matrix has full rank, so the matrix must have linearly independent rows and columns. The number of 1's and 0's must be balanced while maintaining the matrix's invertibility.
Step 2: Constraints and calculation.
For the matrix to be invertible, it must have no rows or columns that are all zeros, which imposes constraints on how the entries can be distributed. By testing different combinations, we find that the minimum difference between the number of 1's and 0's is 1.
Step 3: Conclusion.
Thus, the correct answer is (A).