There are processes that work on a square matrix and "squeez out" a single number. Since matrices in these discussions contain only integers, the process produces a single integer. The evaluation the determinant of a matrix produces a single number. If the matrix is a matrix of coefficients of N linear equations with N unknows, the number is zero if the equations cannot be solved to give the familiar single solution. If the determinant is not zero, then there is that single solution. So the determinant is well-named for this purpose. After further discussion the reader may want to get the program for evaluating determinants. Then click on the link at the beginning of this chapter entitled "program to evaluate a determinant."

But a discussion about the evaluation is needed. Below is a 4x4 matrix (in black letters with subscripts):

                                 1       2       3       4
                       a       a₁      a₂      a₃      a₄
                       b       b₁      b₂      b₃      b₄
                       c       c₁      c₂      c₃      c₄
                       d       d₁      d₂      d₃      d₄

Products will be formed using a b c d in that order, and attaching subscripts. In row a pick any element, say a₃. In the b row pick another element not below a₃, say b₁. In the c pick an element not under either a₃ nor b₁, say c₄. Finally select d₂. The result is the product
                      a₃b₁c₄d₂.
This is one of the 24 products that can be formed from this matrix in the manner that no element picked is under any other picked element.