The matrix rank of a matrix is a fundamental concept in linear algebra that indicates the maximum number of linearly independent rows or columns in the matrix. It provides insight into the dimensionality of the vector space spanned by the rows or columns. A higher rank means more independent information is contained within the matrix. To determine the rank, one can use methods such as row reduction to Row Echelon Form or Reduced Row Echelon Form. The rank can also be related to the solutions of linear equations represented by the matrix, influencing whether a unique solution, no solution, or infinitely many solutions exist.