Matrices and Augmented Form
Translating a system of equations into a compact rectangular array of numbers.
Introduction
Translating a system of equations into a compact rectangular array of numbers.
Past Knowledge
You can solve systems of linear equations using substitution and elimination.
Today's Goal
We represent systems as matrices and learn to read the augmented matrix notation.
Future Success
Matrices are the foundation of linear algebra—used in computer graphics, machine learning, engineering simulations, and data science.
Key Concepts
Definition: Matrix
A matrix is a rectangular array of numbers arranged in rows and columns. An matrix has rows and columns.
Coefficient Matrix
Contains only the coefficients of the variables:
Augmented Matrix
Includes the constants after the vertical bar:
Dimension Notation
We describe matrix size as . A matrix has 3 rows and 4 columns. The augmented matrix above is .
Worked Examples
Example 1: System to Augmented Matrix (Basic)
Write the augmented matrix for:
Step 1: Identify coefficients and constants
Equation 1: coefficients ; constant
Equation 2: coefficients ; constant
Step 2: Arrange in matrix form
Rows = equations, Columns = variables (in order), then constants after the bar
Answer:
Example 2: Three Variables (Intermediate)
Write the augmented matrix for:
Step 1: Align all variables
Order: then constant. Each row is one equation.
Answer:
Example 3: Missing Terms (Advanced)
Write the augmented matrix for:
Key Insight: Missing variables = coefficient 0
Equation 1:
Equation 2:
Equation 3:
Answer:
Common Pitfalls
Forgetting zeros for missing variables
If doesn't appear in an equation, its coefficient is , not blank. Every column must have an entry in every row.
Mixing up rows and columns
Rows = equations, Columns = variables. An matrix means equations in variables (the last column is constants).
Inconsistent variable ordering
Always list variables in the same order (typically alphabetical: ). Reorder terms if needed before writing the matrix.
Real-World Application
Traffic Flow Networks
Civil engineers model intersections as systems of equations where each equation represents conservation of flow: cars entering = cars leaving. With dozens of intersections, matrices make it possible to set up and solve these massive systems computationally.
A city grid with 20 intersections becomes a augmented matrix—far too large to solve by hand, but easily handled by computers using matrix operations.
Practice Quiz
Practice Quiz
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