Number of Matrices of Rank Three in M

To find the number of matrices of rank three in M, we need to understand the properties of rank and how it relates to the number of matrices.

Rank of a Matrix
The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. In other words, it represents the dimension of the vector space spanned by the rows or columns of the matrix.

Rank-1 Matrices in M
A rank-1 matrix is a matrix that can be expressed as the outer product of two vectors. In the context of a 4x3 matrix, a rank-1 matrix can be written as the product of a 4x1 column vector and a 1x3 row vector.

The number of rank-1 matrices in M can be calculated as the product of the number of possible column vectors (3) and the number of possible row vectors (3). Thus, there are 3x3 = 9 rank-1 matrices in M.

Rank-2 Matrices in M
A rank-2 matrix is a matrix that cannot be expressed as the outer product of two vectors, but it can be written as the sum of two rank-1 matrices. In the context of a 4x3 matrix, a rank-2 matrix can be written as the sum of two rank-1 matrices.

To calculate the number of rank-2 matrices in M, we need to consider all possible combinations of two rank-1 matrices. Since there are 9 rank-1 matrices, the number of rank-2 matrices is given by the combination formula C(9, 2).

C(9, 2) = 9! / (2! * (9-2)!) = 9! / (2! * 7!) = 9 * 8 / 2 = 36

Rank-3 Matrices in M
A rank-3 matrix is a matrix that cannot be expressed as the sum of two rank-1 matrices, but it can be written as the sum of three rank-1 matrices. In the context of a 4x3 matrix, a rank-3 matrix can be written as the sum of three rank-1 matrices.

To calculate the number of rank-3 matrices in M, we need to consider all possible combinations of three rank-1 matrices. Since there are 9 rank-1 matrices, the number of rank-3 matrices is given by the combination formula C(9, 3).

C(9, 3) = 9! / (3! * (9-3)!) = 9! / (3! * 6!) = 9 * 8 * 7 / (3 * 2) = 84

Conclusion
In the space M of all 4x3 matrices with entries in the finite field of three elements, the number of matrices of rank three is 84. This can be calculated by considering the combinations of three rank-1 matrices.