![]() ![]() Consider a system of two equations in two variables. To understand Cramer’s Rule, let’s look closely at how we solve systems of linear equations using basic row operations. To find out if the system is inconsistent or dependent, another method, such as elimination, will have to be used. However, if the system has no solution or an infinite number of solutions, this will be indicated by a determinant of zero. Cramer’s Rule is a viable and efficient method for finding solutions to systems with an arbitrary number of unknowns, provided that we have the same number of equations as unknowns.Ĭramer’s Rule will give us the unique solution to a system of equations, if it exists. ![]() Known as Cramer’s Rule, this technique dates back to the middle of the 18th century and is named for its innovator, the Swiss mathematician Gabriel Cramer (1704-1752), who introduced it in 1750 in Introduction à l'Analyse des lignes Courbes algébriques. ![]() We will now introduce a final method for solving systems of equations that uses determinants. Calculating the determinant of a matrix involves following the specific patterns that are outlined in this section.ĭet ( A ) = | 5 2 − 6 3 | = 5 ( 3 ) − ( −6 ) ( 2 ) = 27 det ( A ) = | 5 2 − 6 3 | = 5 ( 3 ) − ( −6 ) ( 2 ) = 27 Using Cramer’s Rule to Solve a System of Two Equations in Two Variables For our purposes, we focus on the determinant as an indication of the invertibility of the matrix. The data can only be decrypted with an invertible matrix and the determinant. Secure signals or messages are sometimes sent encoded in a matrix. Perhaps one of the more interesting applications, however, is their use in cryptography. 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. Evaluating the Determinant of a 2×2 MatrixĪ 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. In this section, we will study two more strategies for solving systems of equations. Some of these methods are easier to apply than others and are more appropriate in certain situations. 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.
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