Modular Matrices

My research will be on one of the many facets of matrix manipulation seen in linear algebra. All matrices have a rank, which is the amount of independent vectors that make up its column or row space. A vector is independent if it a not a linear combination of previous vectors or in simpler terms there are no combination of other vectors in the matrix that can be added together to equal this independent vector. A quantifiable rank for a single can be obtained after some minor calculations and manipulations. More specifically I will see what happens when a given n x n matrix, has all of its values reduced to the entry mod(n). To find the rank of a single matrix as stated is just a matter of some minor calculations, but the description of all of the possible different n x n matrices for any n is significantly more complex. When the modular operation occurs, it causes changes in nearly every property of the matrix including but not limited to rank, the determinant of sub matrices, coefficients of the characteristic polynomial, and the eigenvalues. These properties appear to be described by an equation of group of equations. These equations will be the focal point of my research and will be specifically focused on determining the rank of all square matrix reduced mod(n) for any n.