ABSTRACT
A key theorem, which is non-trivial to prove, is that the row rank and the column rank of a matrix are equal. It is clear that it is not always easy to determine the rank of a matrix, nor even whether it is of full rank, using elementary definitions. In particular, to determine whether a square n × n matrix is of full rank, one can evaluate its determinant (using det(X) in R) and the result that the determinant is non-zero if and only if X is of full rank. To find the exact rank of any symmetric square matrix the result that the rank is equal to the number of non-zero eigenvalues is useful and this can be easily checked in R with the function eigen(X). Some downloadable R packages contain routines for calculating the rank of any matrix.
