Systems of Linear Equations: Determinants and Cramer's Rule

Introduction

This section explores the use of determinants in solving systems of linear equations, focusing on Cramer's Rule. Determinants provide a systematic way to solve systems with as many equations as variables, especially for two or three variables. The section also covers properties of determinants and their applications.

Determinants

2 by 2 Determinants

A 2 by 2 determinant is a special number associated with a 2x2 matrix. It is denoted using vertical bars, not brackets.

• Definition: For real numbers a, b, c, and d, the determinant is written as:

• Value: The value of the determinant is:

• Notation: Use vertical bars for determinants, brackets for matrices.

• Illustration: Multiply diagonally from top left to bottom right, then subtract the product of the other diagonal.

• Example: Evaluate

Cramer's Rule for Two Equations

Solving Systems Using Determinants

Cramer's Rule provides a method for solving a system of two linear equations in two variables using determinants.

• System:

• Determinant of coefficients:

• Determinants for variables:

• Solution:

• Condition: Cramer's Rule applies only if .

• If : The system is either inconsistent (no solution) or dependent (infinitely many solutions). Use substitution or elimination to determine which.

• Example: Solve using Cramer's Rule.

3 by 3 Determinants

Definition and Evaluation

A 3 by 3 determinant is associated with a 3x3 matrix. The value is calculated using minors and cofactors.

• General form:

• Expansion by first row:

• Minors: The minor of entry is the determinant formed by deleting the th row and th column.

• Cofactor: The cofactor is .

• Expansion: To find the value, multiply each entry in a row or column by its cofactor and sum the results. This is called expanding across a row or column.

• Sign pattern: The signs alternate according to , forming a checkerboard pattern starting with + in the top left.

• Example: Evaluate

Cramer's Rule for Three Equations

Solving Systems Using 3 by 3 Determinants

Cramer's Rule can be extended to systems of three equations in three variables.

• System:

• Determinant of coefficients:

• Determinants for variables:

• Solution:

• Condition: Cramer's Rule applies only if .

• If :

  • If at least one of , , or is not zero, the system is inconsistent (no solution).

  • If all are zero, the system is dependent (infinitely many solutions).

• Example: Solve using Cramer's Rule.

Properties of Determinants

Key Theorems

• Theorem 12: Interchanging any two rows (or columns) changes the sign of the determinant.

• Theorem 13: If all entries in any row (or column) are zero, the determinant is zero.

• Theorem 14: If two rows (or columns) are identical, the determinant is zero.

• Theorem 15: If a row (or column) is multiplied by a nonzero constant , the determinant is multiplied by .

• Theorem 16: If a multiple of one row (or column) is added to another row (or column), the value of the determinant does not change.

Examples

• Interchanging Rows: Swapping rows:

• Identical Rows:

• Multiplying a Row: Multiply row 1 by :

• Row Addition: Multiply row 2 by and add to row 1: New row 1: ,

Summary Table: Properties of Determinants

Conclusion

Determinants and Cramer's Rule provide powerful tools for solving systems of linear equations, especially for small systems. Understanding the properties of determinants is essential for efficient computation and for recognizing special cases such as dependent or inconsistent systems.