A determinant, signified by two straight lines ||, is a square array of numbers or symbols that has a specific value. For a square matrix, say, A, there exists an associated determinant, |A|, which has elements identical with the corresponding elements of the matrix. When matrices are not square, they do not possess corresponding determinants.

In general, determinants are expressed as shown in Figure 1, in which a_ijs are called elements of the determinant, and the horizontal and vertical lines of elements are called rows and columns, respectively. The sloping line consisting of a_ii elements is called the principal diagonal of the determinant. Sometimes, determinants can be written in a short form, |a_ij|. The n value, which reflects how many n² quantities are enclosed in ||, determines the order of a determinant.

For determinants of third order, that is, n = 3, or three rows of elements, we can...