Determinants

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 Figure 1. Illustration by Hans & Cassidy. Courtesy of Gale Group. Figure 2. Illustration by Hans & Cassidy. Courtesy of Gale Group. Figure 3. Illustration by Hans & Cassidy. Courtesy of Gale Group. Figure 4. Illustration by Hans & Cassidy. Courtesy of Gale Group. 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 evaluate them as illustrated in Figure 2.

By summing the products of terms as indicated by the arrows pointing towards the right-hand side and subtracting the products of terms as indicated by the arrows pointing towards the left-hand side, we can obtain the value of this determinant. The determinant can also be evaluated in terms of second-order determinants (two rows of elements), as in Figures 3(a) or 3(b).

Each of these second-order determinants, multiplied by an element a_ij, is obtained by deleting the ith row and the jth column of elements in the original third-order determinant, and it is called the "minor" of the element a_ij. The minor is further multiplied by (-1)^I+j, which is exactly the way we determine either the "+" or "-" sign for each determinant included in Figures 3 as shown, to become the "cofactor," C_ij, of the corresponding element.

Determinants have a variety of applications in engineering mathematics. Now, let's consider the system of two linear equations with two unknowns x₁ and x₂: a₁₁x₁ + a₁₂x₂ = b₁ and a₂₁x₁ + a₂₂x₂ = b₂.

We can multiply these two equations by a₂₂ and -a₁₂, respectively, and add them together. This yields (a₁₁a₂₂ - a₁₂a₂₁)x₁ = b₁a₂₂ - b₂a₁₂, i.e., x₁ = (b₁a₂₂ - b₂a₁₂)/(a₁₁a₂₂ a₁₂a₂₁). Similarly, x₂ = (b₁a₂₁ - b₂a₁₁)/(a₁₂a₂₁ - a₁₁a₂₁) can be obtained by adding together the first equation multiplied by a₂₁ and second equation multiplied by -a₁₁. These results can be written in determinant form as in Figure 4.

This is generally called Cramer's rule. Notice that in Figure 4, elements of the determinant in the denominator are the same as the coefficients of x₁ and x₂ in the two equations. To solve for x₁ (or x₂), we then replace the elements that correspond to the coefficients of x₁ (or x₂) of the determinant in the numerator with two constant terms, b₁ and b₂. When b₁ and b₂ both are equal to zero, the system defined by the two equations is said to be homogeneous. In this case, it will have either only the trivial solution x₁ = 0 and x₂ = 0 or additional solutions if the determinant in the denominator in figure 5 is zero. When at least b₁ or b₂ is not zero (that is, a nonhomogeneous system) and the denominator has a value other than zero, the solution to the system is then obtained from figure 4. Cramer's rule is also applicable to systems of three linear equations. Therefore, determinants, along with matrices, have been used for solving simultaneous linear and differential equations involved in various systems, such as reactions in chemical reactors, stiffness of spring-connected masses, and currents in an electric network.