Fat, Square and Thin Matrices - Number of Solutions to Systems of Linear Equations

When a system of equations Ax = b has at least one solution, we say that the system is consistent, otherwise inconsistent. We are concerned with determining when a system of equations is consistent and studying the number of solutions of a system of equations. After having learned the theory of linear algebra, one can observe that the existence (of at least one) and the uniqueness of solutions to a system of equations depends on the rank of the matrix (which is the number of pivot columns of the matrix, and which is also equal to the number of nonzero rows in any echelon form of the matrix), which in turn also depends to some extent on the size of the matrix A (fat, square or thin). For the sake of clarity, let us recall the