Solving Systems of Linear Equations Using Matrices

Solving Systems Using Matrices

Introduction to Matrices in Systems of Equations

Matrices provide a systematic and efficient way to solve systems of linear equations, especially as the number of variables increases. This approach is foundational in algebra and extends to applications in science, engineering, and economics.

• Matrix: A rectangular array of numbers arranged in rows and columns.

• System of Linear Equations: A set of equations with multiple variables, each equation linear in all variables.

Writing the Augmented Matrix of a System

To solve a system using matrices, we first convert the system into an augmented matrix by extracting the coefficients and constants from each equation.

• Write coefficients of each variable in columns.

• Draw a vertical line to separate coefficients from constants (right-hand side values).

• Missing variables in an equation are represented with a coefficient of 0.

Example: Write the augmented matrix for the system:

• 3x + 4y = 7

• 4x - 2y = 5

Augmented matrix:

$\begin{bmatrix} 3 & 4 & | & 7 \\ 4 & -2 & | & 5 \end{bmatrix}$

Writing a System from an Augmented Matrix

Given an augmented matrix, you can reconstruct the original system by assigning each column to a variable and the last column to the constants.

Example: Given the matrix:

$\begin{bmatrix} 1 & 3 & 5 & | & 2 \\ 2 & 5 & 4 & | & 5 \\ 3 & 5 & 4 & | & 6 \end{bmatrix}$

The system is:

• x + 3y + 5z = 2

• 2x + 5y + 4z = 5

• 3x + 5y + 4z = 6

Elementary Row Operations

To solve systems, we perform row operations on the augmented matrix:

• Interchange two rows: $R_i \leftrightarrow R_j$

• Multiply a row by a nonzero constant: $cR_i$

• Add a multiple of one row to another: $R_j + cR_i \rightarrow R_j$

These operations correspond to valid manipulations of the original equations and do not change the solution set.

Row-Echelon Form

A matrix is in row-echelon form if:

• The first nonzero entry in each row (the leading 1) is to the right of the leading 1 in the row above.

• All rows consisting entirely of zeros are at the bottom.

• Each leading 1 is the only nonzero entry in its column below it.

Example of row-echelon form:

$\begin{bmatrix} 1 & * & * & * \\ 0 & 1 & * & * \\ 0 & 0 & 1 & * \end{bmatrix}$

Gaussian Elimination

Gaussian elimination is a systematic method for transforming an augmented matrix into row-echelon form using row operations. Once in this form, the system can be solved by back-substitution.

1. Obtain a leading 1 in the first row, first column (swap or scale rows if necessary).

2. Use row operations to create zeros below this leading 1.

3. Move to the next row and repeat for the next column.

4. Continue until the matrix is in row-echelon form.

5. Solve for variables starting from the bottom row (back-substitution).

Example: Solving a 2x2 System

Solve the system:

• 2x + 3y = 6

• x - 2y = 1

Step 1: Write the augmented matrix:

$\begin{bmatrix} 2 & 3 & | & 6 \\ 1 & -2 & | & 1 \end{bmatrix}$

Step 2: Make the leading entry in row 1 a 1 (swap rows if needed):

$\begin{bmatrix} 1 & -2 & | & 1 \\ 2 & 3 & | & 6 \end{bmatrix}$

Step 3: Eliminate below (row 2 - 2*row 1):

$\begin{bmatrix} 1 & -2 & | & 1 \\ 0 & 7 & | & 4 \end{bmatrix}$

Step 4: Make the leading entry in row 2 a 1 (divide row 2 by 7):

$\begin{bmatrix} 1 & -2 & | & 1 \\ 0 & 1 & | & \frac{4}{7} \end{bmatrix}$

Step 5: Back-substitute to solve for x and y:

• From row 2: $y = \frac{4}{7}$

• From row 1: $x - 2y = 1 \implies x = 1 + 2y = 1 + 2\left(\frac{4}{7}\right) = \frac{15}{7}$

Solution: $x = \frac{15}{7}$, $y = \frac{4}{7}$

Types of Solutions

• Unique Solution: The system reduces to a form where each variable has a unique value.

• No Solution (Inconsistent): The system reduces to a contradiction (e.g., $0 = 4$).

• Infinitely Many Solutions (Dependent): The system reduces to an identity (e.g., $0 = 0$) and at least one variable is free.

Example: Dependent System

Given:

• 3x + 4y = 12

• 6x + 8y = 24

Augmented matrix:

$\begin{bmatrix} 3 & 4 & | & 12 \\ 6 & 8 & | & 24 \end{bmatrix}$

Row 2 - 2*Row 1 gives a row of zeros, indicating infinitely many solutions.

Solving 3x3 Systems

The same principles apply to larger systems. The goal is to use row operations to create zeros below the main diagonal and then back-substitute.

Example: Solve the system:

• x + y + z = 8

• 2x - y + 3z = 9

• x - 2y - z = 2

Augmented matrix:

$\begin{bmatrix} 1 & 1 & 1 & | & 8 \\ 2 & -1 & 3 & | & 9 \\ 1 & -2 & -1 & | & 2 \end{bmatrix}$

Apply Gaussian elimination to reach row-echelon form, then back-substitute to find the solution.

Application Example: Investment Portfolio

Systems of equations can model real-world problems, such as investment allocation.

• Let x = amount in U.S. stocks, y = amount in international stocks, z = amount in bonds.

• Total investment: $x + y + z = 200,000$

• Stocks are four times bonds: $x + y = 4z$

• Expected return: $0.083x + 0.054y + 0.044z = 13,300$

Write as an augmented matrix and solve using Gaussian elimination to find the allocation.

Summary Table: Row Operations

Operation	Notation	Description
Interchange rows	$R_i \leftrightarrow R_j$	Swap two rows
Multiply row by constant	$cR_i$	Multiply all entries in a row by a nonzero constant
Add multiple of one row to another	$R_j + cR_i \rightarrow R_j$	Add a multiple of one row to another row

Practice Problems

• Write the augmented matrix for the system: $8x - 37y = 8$, $2x + 12y = 3$

• Write the system of equations for the matrix: $\begin{bmatrix} -2 & 5 & | & 6 \\ -18 & 5 & | & 26 \end{bmatrix}$

• Solve by Gaussian elimination: $2x + 3y = 12$, $4x + y = 14$

Key Takeaways

• Matrices and row operations provide a procedural method for solving systems of equations.

• Gaussian elimination is a step-by-step process to reach row-echelon form and solve for variables.

• Systems can have one solution, no solution, or infinitely many solutions, depending on the row-reduced form.

• These methods extend to larger systems and real-world applications.

Additional info: This summary includes expanded explanations, step-by-step examples, and a summary table for row operations to ensure the notes are self-contained and suitable for exam preparation.