Representing Systems of Linear Equations using Matrices

Study Guide

Key Definition

A matrix is a set of numbers arranged in rows and columns, used to represent systems of linear equations. For example, the system of equations $2x + 3y = 5$ and $4x - y = 1$ can be represented as follows: Coefficient matrix: $\begin{bmatrix}2 & 3\\4 & -1\end{bmatrix}$ Constant vector: $\begin{bmatrix}5\\1\end{bmatrix}$ Augmented matrix: $\left[\begin{array}{cc|c}2 & 3 & 5\\4 & -1 & 1\end{array}\right]$ Matrix equation: $A\mathbf{x} = \mathbf{b}$ where $A = \begin{bmatrix}2 & 3\\4 & -1\end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix}5\\1\end{bmatrix}$.

Important Notes

A matrix is a rectangular array of numbers or expressions.
Each number in a matrix is called an element.
Matrices are often used to solve systems of linear equations.
Matrices can be added, subtracted, and multiplied.
Understanding matrices is fundamental for higher-level math courses.

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

$\frac{a}{b}$ represents a fraction

$\sqrt{x}$ represents the square root

$x^2$ represents x squared

$[A]$ represents a matrix enclosed in brackets

$|$ is the augmented bar separating coefficients and constants

$R_1 \leftrightarrow R_2$ denotes swapping row 1 and row 2

$kR_1 + R_2 \to R_2$ denotes adding k times row 1 to row 2

Remember to use proper notation when solving problems

Why It Works

Matrices simplify the process of solving systems of linear equations by organizing the coefficients into a single entity, allowing for easier manipulation and solution using operations such as row reduction.

Remember

Matrices are powerful tools in math that simplify complex problems into manageable formats.

Quick Reference

Matrix:$\begin{bmatrix} a & b \\ c & d \end{bmatrix}$

Understanding Representing Systems of Linear Equations using Matrices

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Video explanation of this concept

Beginner

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Beginner Explanation

To form the coefficient matrix, extract the coefficients of each variable from each equation and arrange them in a rectangular grid. To form the constant matrix, list the constants in a column. Then form the augmented matrix by placing a vertical bar between the coefficient columns and the constants. For example, from 2x + 3y = 5 and 4x − y = 1, we get coefficient matrix $\begin{bmatrix}2 & 3\\4 & -1\end{bmatrix}$, constant matrix $\begin{bmatrix}5\\1\end{bmatrix}$, and augmented matrix $\left[\begin{array}{cc|c}2 & 3 & 5\\4 & -1 & 1\end{array}\right]$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Which of the following represents the system of equations $2x + 3y = 7$ and $4x - 5y = 3$ in matrix form?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you are organizing a school event and need to solve for the number of adult and student tickets sold, represented by the equations $3a + 2s = 100$ and $5a + 7s = 200$.

Thinking Challenge

Thinking Exercise