Glossary

9.8 Solving Systems with Cramer's Rule

3 min read•june 25, 2024

Determinants are powerful tools in algebra. They help us solve systems of equations, find areas and volumes, and determine if matrices are invertible. Understanding how to calculate and use determinants is key to mastering linear algebra.

is a method for solving systems of equations using determinants. While it's not always the most efficient approach, it's useful for small systems and provides a direct formula for finding solutions in terms of determinants.

Determinants and Their Properties

Calculation of matrix determinants

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  • of a 2x2 matrix [abcd]\begin{bmatrix}a & b \\ c & d\end{bmatrix} calculated as ad−bcad - bc
  • Determinant of a 3x3 matrix [abcdefghi]\begin{bmatrix}a & b & c \\ d & e & f \\ g & h & i\end{bmatrix} calculated as a(ei−fh)−b(di−fg)+c(dh−eg)a(ei - fh) - b(di - fg) + c(dh - eg)
    • Remember using "" or ""
      • Multiply elements along main diagonal and two parallel diagonals going downwards
      • Subtract products of elements along two parallel diagonals going upwards

Properties of determinants

  • Scalar values associated with square matrices
  • Interchanging any two rows or columns changes sign of determinant
  • Multiplying a row or column by scalar kk multiplies determinant by kk
  • Adding multiple of one row or column to another does not change determinant
  • Determinant of matrix equal to determinant of its transpose
  • Determinant of product of matrices equal to product of their determinants
  • Used to determine if matrix is invertible (non-zero determinant)
  • Calculate area or volume of and
  • Solve systems of linear equations using Cramer's Rule
  • Determinant can be calculated using cofactors and minors of the matrix

Solving Systems with Cramer's Rule

Application of Cramer's Rule

  • For system of two equations {a1x+b1y=c1a2x+b2y=c2\begin{cases}a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2\end{cases}:
    • x=∣c1b1c2b2∣∣a1b1a2b2∣x = \frac{\begin{vmatrix}c_1 & b_1 \\ c_2 & b_2\end{vmatrix}}{\begin{vmatrix}a_1 & b_1 \\ a_2 & b_2\end{vmatrix}} and y=∣a1c1a2c2∣∣a1b1a2b2∣y = \frac{\begin{vmatrix}a_1 & c_1 \\ a_2 & c_2\end{vmatrix}}{\begin{vmatrix}a_1 & b_1 \\ a_2 & b_2\end{vmatrix}}
  • For system of three equations {a1x+b1y+c1z=d1a2x+b2y+c2z=d2a3x+b3y+c3z=d3\begin{cases}a_1x + b_1y + c_1z = d_1 \\ a_2x + b_2y + c_2z = d_2 \\ a_3x + b_3y + c_3z = d_3\end{cases}:
    • x=∣d1b1c1d2b2c2d3b3c3∣∣a1b1c1a2b2c2a3b3c3∣x = \frac{\begin{vmatrix}d_1 & b_1 & c_1 \\ d_2 & b_2 & c_2 \\ d_3 & b_3 & c_3\end{vmatrix}}{\begin{vmatrix}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{vmatrix}}, y=∣a1d1c1a2d2c2a3d3c3∣∣a1b1c1a2b2c2a3b3c3∣y = \frac{\begin{vmatrix}a_1 & d_1 & c_1 \\ a_2 & d_2 & c_2 \\ a_3 & d_3 & c_3\end{vmatrix}}{\begin{vmatrix}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{vmatrix}}, and z=∣a1b1d1a2b2d2a3b3d3∣∣a1b1c1a2b2c2a3b3c3∣z = \frac{\begin{vmatrix}a_1 & b_1 & d_1 \\ a_2 & b_2 & d_2 \\ a_3 & b_3 & d_3\end{vmatrix}}{\begin{vmatrix}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{vmatrix}}

Cramer's Rule vs other methods

  • Less efficient than or for large systems
    • Computational complexity grows factorially with size of system
  • Useful for systems with few equations or when only one variable needs solving
  • Gaussian elimination more efficient for larger systems or when all variables need finding

Real-world interpretation of solutions

  • Represent values of unknown variables that satisfy all equations simultaneously
  • In geometry, can represent coordinates of intersection points or dimensions of shapes
  • In physics, can represent equilibrium positions, velocities, or forces
  • In economics, can represent prices, quantities, or investments balancing supply and demand

Systems of Equations and Their Properties

Types of systems

  • : A set of equations that must be solved together
  • : A system with at least one solution
  • : A system with no solution

Matrix representation

  • Systems of equations can be represented using matrices
  • : Contains coefficients of variables in the system
  • Augmented matrix: Includes the constant terms along with the coefficient matrix

Key Terms to Review (24)

Coefficient Matrix: The coefficient matrix, also known as the system matrix, is a fundamental concept in linear algebra that represents the coefficients of the variables in a system of linear equations. It plays a crucial role in the analysis and solution of such systems, as well as in various applications of matrices and linear transformations.
Cofactor: A cofactor is a non-protein chemical compound or metallic ion that is required for an enzyme's biological activity. It is an essential component that helps facilitate and enhance the catalytic function of an enzyme, enabling it to carry out its specific chemical reactions within the body.
Consistent System: A consistent system is a system of linear equations where there exists at least one solution that satisfies all the equations in the system. In other words, the equations in the system are compatible and have a common solution.
Cramer's Rule: Cramer's rule is a method used to solve systems of linear equations by expressing the solution as a ratio of determinants. It provides a systematic way to find the unique solution to a system of linear equations with the same number of variables and equations.
Determinant: The determinant of a square matrix is a scalar value that is a function of the entries of the matrix. It carries important information about the matrix, such as whether the matrix is invertible and the volume of the parallelepiped spanned by the column vectors of the matrix.
Gabriel Cramer: Gabriel Cramer was a Swiss mathematician who is best known for his work in the field of linear algebra, particularly for developing a method to solve systems of linear equations known as Cramer's rule. This rule provides a way to express the solution of a system of linear equations in terms of the coefficients and constants of the system.
Gaussian Elimination: Gaussian Elimination is a method used to solve systems of linear equations by transforming the system into an equivalent system that is easier to solve. It involves a series of row operations to reduce the system of equations into an upper triangular form, allowing for the systematic solution of the variables.
Inconsistent System: An inconsistent system is a system of linear equations that has no solution, meaning there are no values for the variables that satisfy all the equations in the system simultaneously. This concept is crucial in understanding the behavior of systems of linear equations and their graphical representations.
Linear System: A linear system is a collection of linear equations that describe a relationship between multiple variables. It is a fundamental concept in linear algebra and is widely used in various fields, including mathematics, physics, engineering, and economics.
Matrix: A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns, that can be used to represent and manipulate mathematical relationships and data. Matrices are fundamental tools in linear algebra and have applications in various fields, including physics, engineering, and computer science.
Matrix Inversion: Matrix inversion is the process of finding the inverse of a square matrix, which is a matrix that, when multiplied by the original matrix, results in the identity matrix. The inverse of a matrix is a fundamental concept in linear algebra and is crucial for solving systems of linear equations using techniques like Cramer's Rule.
Minor: In the context of solving systems with Cramer's Rule, a minor is a determinant formed by deleting a row and a column from a larger determinant. Minors play a crucial role in the application of Cramer's Rule, which is a method for solving systems of linear equations by using determinants.
Nonlinear System: A nonlinear system is a mathematical model in which the variables do not have a direct, proportional relationship. This means that the output of the system is not directly proportional to the input, and the system's behavior cannot be predicted by simply analyzing its individual components. Nonlinear systems are commonly encountered in various fields, including physics, engineering, and economics.
Parallelepipeds: A parallelepiped is a three-dimensional geometric shape that has six rectangular faces. It is a generalization of the rectangular prism, where the faces do not have to be squares, but can be any type of rectangle. Parallelepipeds are an important concept in the context of solving systems of linear equations using Cramer's rule.
Parallelograms: A parallelogram is a quadrilateral with two pairs of parallel sides. These four-sided shapes have several distinctive properties that make them useful in various mathematical contexts, including the topic of solving systems with Cramer's Rule.
Row Reduction: Row reduction is a fundamental matrix operation used to simplify and solve systems of linear equations. It involves performing a series of elementary row operations on a matrix to transform it into an equivalent matrix in row echelon form, which can then be used to determine the solution to the system.
Rule of Sarrus: The Rule of Sarrus is a method used to calculate the determinant of a 3x3 matrix. It provides a systematic way to expand the determinant along the first row or column, making it a useful tool in solving systems of linear equations using Cramer's Rule.
Simultaneous Equations: Simultaneous equations are a set of two or more equations that share common variables and must be solved together to find the values of those variables. They are a fundamental concept in algebra and linear equations, allowing for the analysis of complex systems with multiple unknowns.
System of Equations: A system of equations is a set of two or more equations that share common variables and must be solved simultaneously to find the values of those variables. It is a fundamental concept in mathematics, particularly in the context of linear algebra and solving for unknown quantities.
Triangle Method: The Triangle Method is a technique used to solve systems of linear equations by representing the system geometrically as a triangle. This method allows for the visualization and intuitive understanding of the relationships between the variables and the solution of the system.
Uniqueness Theorem: The Uniqueness Theorem is a fundamental principle in mathematics that guarantees the existence and exclusivity of a solution to a system of linear equations. It ensures that for a given system of linear equations, there is at most one unique solution that satisfies all the equations simultaneously.
Δ (Delta): Δ, also known as the delta symbol, is a mathematical symbol used to represent change or difference. It is commonly employed in various mathematical and scientific contexts, including the field of systems of linear equations and the application of Cramer's rule for solving such systems.
Δx: Δx, also known as the delta-x or change in x, is a fundamental concept in calculus and mathematical analysis. It represents the change or difference in the independent variable x between two points or values. This term is particularly important in the context of solving systems of linear equations using Cramer's rule, as it is a crucial component in the calculation of the determinant and the solution of the system.
Δy: Δy, or delta y, represents the change in the dependent variable y with respect to a change in the independent variable. It is a fundamental concept in calculus and is used to quantify the rate of change of a function at a specific point or over an interval.
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