@MISC{MATRIX13.MIZ,
  AUTHOR = {P\c{a}k, Karol},
  TITLE = {Basic Properties of the Rank of Matrices
           over a Field},
  SECTION1 = {Triangular Matrices},
  SECTION2 = {The Rank of Matrices},
  DAY = {28},
  MONTH = {August},
  YEAR = {2007},
  ADDRESS1 = {University of Bialystok, ul. Akademicka 2,
              15-267 Bialystok},
  SUMMARY = {In this paper we present selected properties
             of triangular matrices and basic properties of
             the rank of matrices over a field.\par
             We define a submatrix as a matrix formed by
             selecting certain rows and columns from a bigger
             matrix. That is in our considerations, as an array,
             it is cut down to those entries constrained by row
             and column. Then we introduce the concept of the rank
             of a $m\times n$ matrix $A$ by the condition:
             $A$ has the rank $r$ if and only if,
             there is a $r\times r$ submatrix of $A$ with
             a non-zero determinant, and for every $k\times k$
             submatrix of $A$ with a non-zero determinant we have $k\leq r$.\par
             At the end, we prove that the rank defined by the size
             of the biggest submatrix with a non-zero determinant of a matrix $A$,
             is the same as the maximal number of linearly independent
             rows of $A$.}}