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 array of real numbers. It is denoted using vertical bars, distinguishing it from a matrix, which uses 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.

• Example: Evaluate :

Cramer's Rule for Two Equations

Solving a System of Two Linear Equations

Cramer's Rule provides a method to solve a system of two linear equations using determinants, provided the system has a unique solution.

• General System:

• Determinants:

• Solution (Cramer's Rule):

• Condition: (the system has a unique solution).

• 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 array of real numbers. The value is calculated using minors and cofactors.

• General Form:

• Expansion by Minors: The value is:

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

• Cofactor: The cofactor is .

• Expansion: The determinant can be expanded along any row or column, multiplying each entry by its cofactor and summing the results.

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

• Example: Evaluate :

Cramer's Rule for Three Equations

Solving a System of Three Linear Equations

Cramer's Rule can be extended to solve a system of three equations in three variables, provided the determinant of the coefficient matrix is nonzero.

• General System:

• Determinants:

• Solution (Cramer's Rule):

• Condition: (unique solution exists).

• 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) of a determinant changes its sign.

• 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: Adding a multiple of one row (or column) to another row (or column) does not change the value of the determinant.

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: , (unchanged)

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 when unique solutions exist.