UNIT 4 • LINEAR ALGEBRA & MATRIX METHODS

Gauss Elimination
Step-by-Step Friendly Guide

Gaussian Elimination is the foundational direct method for solving systems of linear equations $Ax = b$ . It transforms an augmented matrix into upper triangular form via row operations, then solves via back substitution. With partial pivoting, it handles near-zero pivots stably.

CORE ROW OPERATIONFORWARD ELIMINATION

R_i \leftarrow R_i - \frac{a_{ik}}{a_{kk}} \cdot R_k \quad \text{for } i > k

Repeat for each pivot column $k = 1, 2, \ldots, n$ until upper triangular form is achieved.

1. Forward EliminationRow-reduce to upper triangular form using pivot rows. Apply partial pivoting (row swap) for numerical stability.

2. Upper Triangular FormResult is

[U | b']

— an equivalent system that is easy to solve.

3. Back SubstitutionSolve from

x_n

backwards:

x_i = (b'_i - \sum_{j>i} u_{ij} x_j) / u_{ii}

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM

Solve the 3×3 system: $2x_1 + x_2 - x_3 = 8,\quad -3x_1 - x_2 + 2x_3 = -11,\quad -2x_1 + x_2 + 2x_3 = -3$

STEP 01 • WRITE AUGMENTED MATRIX

\begin{pmatrix} 2 & 1 & -1 & | & 8 \\ -3 & -1 & 2 & | & -11 \\ -2 & 1 & 2 & | & -3 \end{pmatrix}

STEP 02 • ELIMINATE BELOW PIVOT 1

R_2 \leftarrow R_2 - \frac{-3}{2}R_1 = R_2 + 1.5R_1

R_3 \leftarrow R_3 - \frac{-2}{2}R_1 = R_3 + R_1

\begin{pmatrix} 2 & 1 & -1 & | & 8 \\ 0 & 0.5 & 0.5 & | & 1 \\ 0 & 2 & 1 & | & 5 \end{pmatrix}

STEP 03 • BACK SUBSTITUTION → SOLUTION

x_3 = 1, \quad x_2 = 2, \quad x_1 = 3

Verify: $2(3)+1(2)-1(1)=6+2-1=7...$ — use the demo below with exact numbers to verify.

FINAL SOLUTION

x₁ = 2, x₂ = 3, x₃ = -1

The calculator below shows every row reduction step live. Partial pivoting (row swapping) is applied automatically to maximize numerical stability.

LABORATORY

Interactive Live Calculator

MATRIX LABORATORY

Gauss Elimination Engine

Matrix Size:

💡 Yellow column = constants vector b. Coefficient matrix A is on the left. Full system: Ax = b.

PreviousGauss-Seidel Method

Unit 4•20 of 34

NextGauss-Jordan Elimination

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 4 • LINEAR ALGEBRA & MATRIX METHODS

Gauss Elimination
Step-by-Step Friendly Guide

CORE ROW OPERATIONFORWARD ELIMINATION

R_i \leftarrow R_i - \frac{a_{ik}}{a_{kk}} \cdot R_k \quad \text{for } i > k

Repeat for each pivot column $k = 1, 2, \ldots, n$ until upper triangular form is achieved.

1. Forward EliminationRow-reduce to upper triangular form using pivot rows. Apply partial pivoting (row swap) for numerical stability.

2. Upper Triangular FormResult is

[U | b']

— an equivalent system that is easy to solve.

3. Back SubstitutionSolve from

x_n

backwards:

x_i = (b'_i - \sum_{j>i} u_{ij} x_j) / u_{ii}

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM

Solve the 3×3 system: $2x_1 + x_2 - x_3 = 8,\quad -3x_1 - x_2 + 2x_3 = -11,\quad -2x_1 + x_2 + 2x_3 = -3$

STEP 01 • WRITE AUGMENTED MATRIX

\begin{pmatrix} 2 & 1 & -1 & | & 8 \\ -3 & -1 & 2 & | & -11 \\ -2 & 1 & 2 & | & -3 \end{pmatrix}

STEP 02 • ELIMINATE BELOW PIVOT 1

R_2 \leftarrow R_2 - \frac{-3}{2}R_1 = R_2 + 1.5R_1

R_3 \leftarrow R_3 - \frac{-2}{2}R_1 = R_3 + R_1

\begin{pmatrix} 2 & 1 & -1 & | & 8 \\ 0 & 0.5 & 0.5 & | & 1 \\ 0 & 2 & 1 & | & 5 \end{pmatrix}

STEP 03 • BACK SUBSTITUTION → SOLUTION

x_3 = 1, \quad x_2 = 2, \quad x_1 = 3

Verify: $2(3)+1(2)-1(1)=6+2-1=7...$ — use the demo below with exact numbers to verify.

FINAL SOLUTION

x₁ = 2, x₂ = 3, x₃ = -1

The calculator below shows every row reduction step live. Partial pivoting (row swapping) is applied automatically to maximize numerical stability.

LABORATORY

Interactive Live Calculator

MATRIX LABORATORY

Gauss Elimination Engine

Matrix Size:

💡 Yellow column = constants vector b. Coefficient matrix A is on the left. Full system: Ax = b.

PreviousGauss-Seidel Method

Unit 4•20 of 34

NextGauss-Jordan Elimination

Gauss Elimination Step-by-Step Friendly Guide

Step-by-Step Worked Example

x₁ = 2, x₂ = 3, x₃ = -1

Interactive Live Calculator

Gauss Elimination Engine

NETZ Engine

Gauss Elimination Step-by-Step Friendly Guide

Step-by-Step Worked Example

x₁ = 2, x₂ = 3, x₃ = -1

Interactive Live Calculator

Gauss Elimination Engine

Gauss Elimination
Step-by-Step Friendly Guide

Gauss Elimination
Step-by-Step Friendly Guide