# 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_{ij}s 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 n^{2} 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 *i*th row and the *j*th 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 x_{1} and x_{2}: a_{11}x_{1 }+ a_{12}x_{2} = b_{1} and a_{21}x_{1} + a_{22}x_{2} = b_{2}.

We can multiply these two equations by a_{22} and -a_{12}, respectively, and add them together. This yields (a_{11}a_{22} - a_{12}a_{21})x_{1} = b_{1}a_{22} - b_{2}a_{12}, i.e., x_{1} = (b_{1}a_{22} - b_{2}a_{12})/(a_{11}a_{22} a_{12}a_{21}). Similarly, x_{2} = (b_{1}a_{21} - b_{2}a_{11})/(a_{12}a_{21} - a_{11}a_{21}) can be obtained by adding together the first equation multiplied by a_{21} and second equation multiplied by -a_{11}. 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 x_{1} and x_{2} in the two equations. To solve for x_{1} (or x_{2}), we then replace the elements that correspond to the coefficients of x_{1} (or x_{2}) of the determinant in the numerator with two constant terms, b_{1} and b_{2}. When b_{1} and b_{2} 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 x_{1} = 0 and x_{2} = 0 or additional solutions if the determinant in the denominator in figure 5 is zero. When at least b_{1} or b_{2} 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.

Pang-Jen Kung

## Additional topics

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