In this paper I present selected properties of triangular matrices and basic properties of the rank of matrices over a field.

I define a submatrix as a matrix formed by selecting certain rows and columns from a bigger matrix. That is in my considerations, as an array, it is cut down to those entries constrained by row and column. Then I introduce the concept of the rank of a *m x n* matrix *A* by the condition: *A* has the rank *r* if and only if, there is a *r x r* submatrix of *A* with a non-zero determinant, and for every *k x k* submatrix of *A* with a non-zero determinant we have *k ≤ r.*

At the end, I 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.*

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