Chapter 5: Systems of Equations and Matrices – College Algebra Study Notes

Chapter 5: Systems and Matrices

5.1 Systems of Linear Equations

Systems of linear equations are collections of two or more linear equations involving the same set of variables. The solution to a system is the set of variable values that satisfy all equations simultaneously.

• Linear System: A set of equations in which each equation is linear (the highest power of any variable is 1).

• Consistent System: Has at least one solution.

• Inconsistent System: Has no solution.

• Dependent System: Has infinitely many solutions (the equations describe the same line).

• Independent System: Has exactly one solution.

Possible Graphs of Linear Systems in Two Variables

• Intersecting Lines: One solution (the point of intersection). The system is consistent and independent.

• Parallel Lines: No solution. The system is inconsistent.

• Coinciding Lines: Infinitely many solutions. The system is consistent and dependent.

Example Table: Types of Linear Systems

Graph	Number of Solutions	System Type
Intersecting Lines	1	Consistent, Independent
Parallel Lines	0	Inconsistent
Coinciding Lines	Infinitely Many	Consistent, Dependent

Solution to a System

• To check if an ordered pair is a solution, substitute the values into each equation and verify both are true.

• Example: For the system $x + y = 2$, $x - y = 0$, check if (1,1) is a solution:

  • $1 + 1 = 2$ (True)

  • $1 - 1 = 0$ (True)

  Thus, (1,1) is a solution.

Substitution Method

This method involves solving one equation for one variable and substituting into the other equation.

1. Solve one equation for one variable.

2. Substitute this expression into the other equation.

3. Solve for the remaining variable.

4. Substitute back to find the other variable.

Elimination Method (Addition Method)

This method involves adding or subtracting equations to eliminate one variable, making it possible to solve for the other.

1. Multiply one or both equations by suitable numbers so that the coefficients of one variable are opposites.

2. Add or subtract the equations to eliminate one variable.

3. Solve for the remaining variable.

4. Substitute back to find the other variable.

Special Systems

• No Solution: The system is inconsistent (e.g., parallel lines).

• Infinitely Many Solutions: The system is dependent (e.g., same line).

Linear Systems with Three Unknowns (Variables)

To solve a system with three variables, use elimination to reduce the system to two equations in two variables, then solve as before.

1. Eliminate a variable from any two equations.

2. Eliminate the same variable from a different pair of equations.

3. Solve the resulting two-variable system.

4. Back-substitute to find the third variable.

5.2 Matrix Solution of Linear Systems

The Matrix

• A matrix is a rectangular array of numbers arranged in rows and columns.

• An augmented matrix represents a system of equations, including the constants from the right side of the equations.

• The size of a matrix is given as "number of rows × number of columns".

Example Table: Matrix Representation

System of Equations	Augmented Matrix
$x + 2y = 3$ $3x - 4y = -12$	$\begin{bmatrix}1 & 2 & 3 \\ 3 & -4 & -12\end{bmatrix}$

5.3 Determinant Solution of Linear Systems

Determinants

• The determinant of a square matrix is a special number calculated from its elements.

• For a 2×2 matrix $\begin{bmatrix}a & b \\ c & d\end{bmatrix}$, the determinant is $ad - bc$.

• For a 3×3 matrix, the determinant is calculated by expanding along a row or column.

Example: 2×2 Determinant

• Given $\begin{bmatrix}2 & 3 \\ 1 & 4\end{bmatrix}$, determinant is $2 \times 4 - 3 \times 1 = 8 - 3 = 5$.

Cramer's Rule

• Cramer's Rule provides a method to solve a system of linear equations using determinants.

• For a system:

  • $a_1x + b_1y = c_1$

  • $a_2x + b_2y = c_2$

  The solution is:

  • $x = \dfrac{D_x}{D}$

  • $y = \dfrac{D_y}{D}$

  where $D$ is the determinant of the coefficient matrix, $D_x$ and $D_y$ are determinants formed by replacing the respective columns with the constants.

• If $D = 0$, Cramer's Rule does not apply (system is either dependent or inconsistent).

General Form of Cramer's Rule

• For $n$ equations in $n$ variables, $D$ is the determinant of the coefficient matrix.

• $D_i$ is the determinant formed by replacing the $i$th column with the constants.

• If $D \neq 0$, the system has a unique solution: $x_i = \dfrac{D_i}{D}$.

5.5 Nonlinear Systems of Equations

A nonlinear system contains at least one equation that is not linear (e.g., quadratic, circle, etc.).

• Solutions are found by substitution or elimination, similar to linear systems.

• Graphical solutions may involve curves intersecting lines or other curves.

• Systems may have real or complex solutions, or no solution.

5.7 Properties of Matrices

Basic Definitions

• A square matrix has the same number of rows and columns.

• A zero matrix contains only zeros.

Matrix Addition and Subtraction

• Matrices of the same size can be added or subtracted by adding or subtracting corresponding elements.

• Example: $\begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix} + \begin{bmatrix}5 & 6 \\ 7 & 8\end{bmatrix} = \begin{bmatrix}6 & 8 \\ 10 & 12\end{bmatrix}$

Scalar Multiplication

• Multiplying a matrix by a scalar means multiplying every element by that number.

• Example: $2 \times \begin{bmatrix}1 & 3 \\ 2 & 4\end{bmatrix} = \begin{bmatrix}2 & 6 \\ 4 & 8\end{bmatrix}$

Matrix Multiplication

• To multiply matrices $A$ (size $m \times n$) and $B$ (size $n \times p$), the number of columns in $A$ must equal the number of rows in $B$.

• The product $AB$ is an $m \times p$ matrix.

• Matrix multiplication is not commutative: $AB \neq BA$ in general.

• Matrix multiplication is associative and distributive over addition.

Example Table: Matrix Multiplication Possibility

Matrix Sizes	Can Multiply?	Resulting Size
2×3 and 3×4	Yes	2×4
3×2 and 2×3	Yes	3×3
2×3 and 2×3	No	—

Properties of Matrix Operations

• Associative Law: $(AB)C = A(BC)$

• Distributive Law: $A(B + C) = AB + AC$

Additional info: These notes cover all major aspects of Chapter 5: Systems of Equations and Matrices, including linear and nonlinear systems, matrix operations, determinants, and Cramer's Rule, as relevant to a College Algebra course.