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Cramer's Rule: Solving Systems of Linear Equations with Determinants

Zang Li^{* *}Correspondence: Zang Li, Department of Mathematics, Soochow University, Jiangsu, China, Email:

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Description

In the realm of linear algebra, solving systems of linear equations is a fundamental task with applications in various fields, including engineering, physics, and economics. Cramer's rule, named after the swiss mathematician Gabriel Cramer, provides an elegant and efficient method for solving such systems by leveraging the concept of determinants. This rule, formulated in the early 18th century, remains an essential tool for solving small-sized systems of equations.

Cramer's rule provides a solution to a system of n linear equations with n variables. Let’s consider a system of equations in matrix form as ax - b, where a is an n× n coefficient matrix, x is a column vector containing the variables, and b is a column vector representing the constants on the right-hand side of the equations. According to cramer's rule, the solution for each variable x _ i can be obtained by dividing the determinant of a modified matrix by the determinant of the coefficient matrix.

To apply cramer's rule, let's assume that the determinant of the coefficient matrix, denoted by |a| , is non-zero. In this case, the system has a unique solution.

The steps for solving the system using cramer's rule

Calculate the determinant of the coefficient matrix a, denoted by |a| . This determinant serves as the denominator for finding the solutions.

For each variable x _ i , create a new matrix by replacing the th i−^th column of a with the column vector b. Let's call this modified matrix a _ i .

Calculate the determinant of each modified matrix a _ i , denoted by |a_i| . Each determinant represents the numerator for the corresponding variable's solution.

Obtain the solution for each variable x _ i by dividing the determinant |a_i| by the determinant |a| .

Cramer's rule provides an explicit expression for each variable, making it particularly useful for solving small systems of equations. However, it is worth noting that cramer's rule becomes computationally expensive as the size of the system increases. This is because the determinants must be calculated for each variable, requiring considerable computational effort and potentially leading to numerical instability.

Furthermore, cramer's rule has limitations when applied to systems with singular or nearly singular coefficient matrices. In such cases, the determinant of the coefficient matrix |a| becomes zero or close to zero, rendering the rule invalid. These situations require alternative methods, such as gaussian elimination or matrix factorization techniques, to solve the system effectively.

Despite its limitations, cramer's rule remains a valuable tool in mathematics and serves as a foundational concept for understanding the relationship between linear equations, determinants, and matrices. It provides insight into the behavior of systems of linear equations and offers a straightforward approach to finding their solutions when applicable.

Moreover, cramer's rule has connections to other areas of mathematics, such as the inverse of a matrix. By applying cramer's rule to the augmented matrix [a|b], one can determine if the system is solvable and, if so, obtain the unique solution. This relationship highlights the broader applicability and importance of cramer's rule within the broader context of linear algebra.

Cramer's rule provides an elegant method for solving systems of linear equations using determinants. By leveraging the power of determinants, cramer's rule offers a straightforward approach to finding solutions for small systems of equations. Although it may not be practical for larger systems or systems with singular matrices, it remains a valuable concept that contributes to the understanding of linear algebra and its applications in various fields.