AUGMENTED MATRIX
What is an augmented matrix?
A matrix that includes both the coefficients and the constants from a system of linear equations.
Name the three types of row operations.
1. Row switching
2. Row multiplication
3. Row addition/subtraction
What is the goal of Gaussian elimination?
To transform a system into an upper triangular matrix to solve using back-substitution.
How is Gauss-Jordan elimination different from Gaussian elimination?
Gauss-Jordan continues past row echelon (zeros under 1s) to reduced row-echelon form (zeros everywhere other than the 1s) with leading 1s and zeros everywhere else in the pivot columns.
Apply Gaussian elimination to the system :
{x + y = 2
{2x + 2y = 5
Augmented Matrix:
[1 1 | 2]
[2 2 | 5]
Use row operation (row 2 = row 2 - (2 x row 1)
[1 1 | 2]
[0 0 | 1]
This means that the second row becomes 0x + 0y = 1
Meaning it has no solution
True or False: An augmented matrix always has one unique solution.
False. It can also have no solution or infinitely many.
What row operation would you use to swap two rows?
Row switching
What form should the matrix be in after Gaussian elimination?
Row echelon form (zeros under 1s).
What is the final form of a matrix after Gauss-Jordan elimination?
Reduced row-echelon form.
What happens if you apply Gaussian elimination to a system with fewer equations than variables?
There will be at least one free variable ⇒ infinite solutions.
Write the augmented matrix for the system:
x + y = 2
2x - 3y = 5
[1 1 | 2]
[2 -3 | 5]
Change Row 2 to --> R2 - 3R1 (Switch Row 2 into Row 2 Minus 3 times Row 1):
[1 2 | 3]
[3 6 | 9]
[1 2 | 3]
[0 0 | 0]
Solve using Gaussian elimination:
x + y + z = 6
2x + 3y + z = 14
x + 2y + 3z = 14
Solution x = 1, y = 2, z = 3
Solve with Gauss-Jordan:
x + y = 4, 2x + 3y = 9
Solution: x = 3, y = 1
A student solves a system describing the cost of pencils and pens:
x + y = 3
2x+2y=6
What does this tell us about the system (in terms of solutions)?
The equations are dependent (multiples of each other) ⇒ infinitely many solutions.
Explain the difference between a coefficient matrix and an augmented matrix.
A coefficient matrix includes only the coefficients of variables, while an augmented matrix includes the constants as a last column.
Why are row operations important when solving matrix systems?
They help transform the matrix to a simpler form to solve the system easily.
How do you know when to stop applying Gaussian elimination?
When the matrix is in upper triangular form (zeros under 1s).
What do you do if a pivot element is zero?
Swap rows to bring a non-zero element into the pivot position.
Apply Gaussian elimination to:
x + 2y + z = 5
2x + 4y + 2x = 10
Second equation is a multiple of the first ⇒ dependent ⇒ reduce to 1 equation in 3 variables ⇒ Infinite solutions with 2 free variables
Convert the system 3x + 2y = 7, x - y = 4 into an augmented matrix.
[3 2 | 7]
[1 -1 | 4]
Use row operations to create a leading 1 in the top-left corner of the matrix.
[0 1 | 4]
[2 3 | 7]
Step 1 is to swap Row 1 and Row 2
Step 2 is to Divide Row 1 by 2
Your final answer should be:
[1 1.5 | 3.5]
[0 1 | 4]
True or False: Gaussian elimination always leads to reduced row-echelon form.
False. Gaussian stops at row echelon (zeros under 1s only).
Apply Gauss-Jordan elimination to:
[1 2 | 5]
[3 4 | 11]
Final result:
[1 0 | 1]
[0 1 | 2]
Soltuion: x = 1, y = 2
True or False: A row of all zeros in the coefficient part always means no solution. Explain.
False – It depends on the constant. If the entire row (including the constant) is 0, it's a dependent equation ⇒ infinite solutions. If constant ≠ 0, it's inconsistent ⇒ no solution.