3.5 Determinants and Cramer’s Rule
We have learned how to solve systems of equations in two variables and three variables, and by multiple methods: substitution, addition, Gaussian elimination, using the inverse of a matrix, and graphing. Some of these methods are easier to apply than others and are more appropriate in certain situations. In this section, we will study two more strategies for solving systems of equations.
Evaluating the Determinant of a 2×2 Matrix
A determinant is a real number that can be very useful in mathematics because it has multiple applications, such as calculating area, volume, and other quantities. Here, we will use determinants to reveal whether a matrix is invertible by using the entries of a square matrix to determine whether there is a solution to the system of equations. Perhaps one of the more interesting applications, however, is their use in cryptography. Secure signals or messages are sometimes sent encoded in a matrix. The data can only be decrypted with an invertible matrix and the determinant. For our purposes, we focus on the determinant as an indication of the invertibility of the matrix. Calculating the determinant of a matrix involves following the specific patterns that are outlined in this section.
Find the Determinant of a 2 2 Matrix
The determinant of 2 2 matrix, given
is defined as:
det
Notice that for determinants, we use straight vertical lines. In other words, det.
Does look familiar? It should. Remember this was the denominator in the scalar we multiply to the 2 2 matrix with the diagonals switched and the off-diagonals opposite to create the inverse. So the inverse of a 2 2 matrix like above would be:
Example Finding the Determinant of a 2 2 Matrix
Find the determinant of the given matrix
det
Using Cramer’s Rule to Solve a System of Two Equations in Two Variables
We will now introduce a final method for solving systems of equations that uses determinants. Known as Cramer’s rule, this technique dates back to the middle 18th century and is named for its innovator, the Swiss mathematician Gabriel Cramer (1704-1752), who introduced it in 1750. Cramer’s Rule is a viable and efficient method for finding solutions to systems with any number of unknowns, provided that we have the same number of equations as unknowns.
Cramer’s Rule will give us the unique solution to a system of equations, if it exists. However, if the system has no solution or an infinite number of solutions, this will be indicated by a determinant, . Inconsistent Solutions have at least one numerator determinant that is non-zero. Dependent solutions have zero as the determinant in both the numerators. Another method must be used to find the general solution.
To see how and why Cramer’s Rule works, we will direct you to the source material at OpenStax College Algebra.
In short, Cramer’s rule begins with a system of equations, such as:
and we can show that
Notice that the denominator for both and is the determinant of the coefficient ma