Let M be the set of all invertible 5 × 5 matrices with entries 0 and 1...
Introduction:
We are given a set M of all invertible 5 × 5 matrices with entries 0 and 1. We need to find the number of 1's and 0's in each matrix of M.
Explanation:
To find the number of 1's and 0's in each matrix of M, we need to understand the properties of invertible matrices and the possible combinations of 0's and 1's in such matrices.
Properties of Invertible Matrices:
- Invertible matrices have a non-zero determinant.
- The determinant of a matrix can be calculated using the cofactor expansion method or by using row operations to transform the matrix into echelon form.
Possible Combinations of 0's and 1's:
- Since the matrices in M are invertible, they must have a non-zero determinant. This implies that the matrices cannot have any row or column consisting entirely of 0's.
- Let's consider the number of 1's and 0's in each row and column of a matrix in M.
- If a row or column has all 1's, the determinant of the matrix will be 0. Therefore, each row and column must have at least one 0.
- Let's assume the number of 1's in a row is a and the number of 0's is b. Similarly, the number of 1's in a column is c and the number of 0's is d.
- Since each row and column must have at least one 0, we have a + b = 5 and c + d = 5.
- Also, the number of 1's and 0's in each row and column must add up to 5. Therefore, a + c = b + d = 5.
Calculating n0(M) and n1(M):
- From the above deductions, we can conclude that there are two possibilities for the number of 1's and 0's in each row and column: (1,4) or (2,3).
- For (1,4), there are 5 choices for the position of 0 in each row and column. Therefore, the total number of matrices with (1,4) combination is 5^5.
- For (2,3), there are 5 choices for the positions of 0's in each row and column. Therefore, the total number of matrices with (2,3) combination is 5^2 * 5^3.
- Hence, n0(M) = 5^5 + 5^2 * 5^3 and n1(M) = 5^5 + 5^2 * 5^3.
Summary:
- The number of 1's and 0's in each matrix of M can be determined by considering the properties of invertible matrices and the possible combinations of 0's and 1's in such matrices.
- From the analysis, we found that n0(M) = 5^5 + 5^2 * 5^3 and n1(M) = 5^5 + 5^2 * 5^3.