Augmented Matrix Calculator - Augmented Matrix Calculator RREF & Linear System Solver
Free augmented matrix calculator & RREF calculator. Solve systems of linear equations, calculate reduced row echelon form & perform Gaussian elimination. Our calculator uses matrix row operations to transform augmented matrices into RREF and extract solutions for consistent and inconsistent systems.
Last updated: December 15, 2024
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Enter coefficients for the system of equations. The augmented column (after |) contains the constants.
Solution
Reduced Row Echelon Form (RREF):
Solution Steps:
- Step 1: Set up the augmented matrix [A|b]
- Step 2: Apply Gaussian elimination to reach row echelon form
- Step 3: Continue to reduced row echelon form (RREF)
- Step 4: System has a unique solution
- Step 5: Read solution from RREF matrix
Matrix Solution Types:
- • Unique: System has exactly one solution (consistent & independent)
- • Infinite: System has infinitely many solutions (consistent & dependent)
- • Inconsistent: System has no solution (parallel lines/planes)
Augmented Matrix Calculator Features & Methods
RREF Properties
Leading 1s, Zero Columns, Identity Form
Transforms matrices to their simplest form for immediate solution reading
System Format
[A | b] → [I | x]
Combines coefficient matrix with constants for efficient solving
Row Operations
Swap, Scale, Add
Uses elementary row operations to reach row echelon form
Echelon Forms
REF → RREF
Progressive simplification to most reduced form
Solution Types
Unique, Infinite, None
Identifies all possible solution scenarios automatically
Step-by-step
Full Work Shown
Educational tool showing every row operation applied
Example System Solution
System: 2x + y - z = 8, -3x - y + 2z = -11, -2x + y + 2z = -3
x =
2
y =
3
z =
-1
How Our Augmented Matrix Calculator Works
Our augmented matrix calculator solves systems of linear equations using Gaussian elimination and row reduction to reach reduced row echelon form (RREF). The calculator applies elementary row operations systematically to transform the augmented matrix [A|b] into a form where solutions can be read directly.
The Gaussian Elimination Process
Step 1: Form augmented matrix [A | b]
Step 2: Create zeros below pivot positions (forward elimination)
Step 3: Scale rows to make pivots equal to 1
Step 4: Create zeros above pivots (back substitution)
Result: RREF form [I | x] where x is the solution
Elementary row operations preserve the solution set while simplifying the matrix structure. The three allowed operations are: swapping two rows, multiplying a row by a non-zero scalar, and adding a multiple of one row to another row.
Shows progression from augmented matrix to RREF
Mathematical Foundation
The augmented matrix method is based on the fundamental theorem that elementary row operations preserve the solution set of a linear system. By systematically applying these operations, we can transform any linear system into an equivalent but simpler form where the solution is immediately apparent.
- Augmented matrix combines coefficients and constants: [A | b]
- Row operations create leading 1s (pivots) in systematic positions
- RREF has identity matrix on left side for unique solutions
- Missing pivots indicate free variables and infinite solutions
- Row [0 0 ... 0 | k] where k ≠ 0 indicates inconsistent system
- Rank of coefficient matrix determines solution type
Sources & References
- Linear Algebra and Its Applications - David C. Lay, Steven R. Lay, Judi J. McDonald (6th Edition)Standard reference for matrix methods and RREF
- Elementary Linear Algebra - Howard Anton, Chris Rorres (11th Edition)Comprehensive coverage of Gaussian elimination
- Khan Academy - Linear Algebra CourseFree educational resources for matrix operations
Need help with other linear algebra calculations? Check out our Gaussian elimination calculator and cross product calculator.
Get Custom Calculator for Your PlatformAugmented Matrix Calculator Examples
Original System:
2x + y - z = 8
-3x - y + 2z = -11
-2x + y + 2z = -3
Augmented Matrix:
[ 2 1 -1 | 8 ]
[ -3 -1 2 | -11]
[ -2 1 2 | -3 ]
Solution Steps:
- Set up augmented matrix [A|b]
- Apply row operations to create pivots
- Eliminate below and above pivots
- Scale to get leading 1s
- Reach RREF: [I | solution]
RREF Result:
[ 1 0 0 | 2 ]
[ 0 1 0 | 3 ]
[ 0 0 1 | -1 ]
Solution: x = 2, y = 3, z = -1
The system has a unique solution (consistent and independent).
Infinite Solutions Example
x + 2y = 4
2x + 4y = 8
RREF shows free variable: y = t, x = 4 - 2t (infinitely many solutions)
No Solution Example
x + y = 2
x + y = 5
RREF shows [0 0 | 3]: inconsistent system (parallel lines)
Frequently Asked Questions
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