Mastering Systems of Linear Equations: Gaussian Elimination and Augmented Matrices

1 System of Equations

A linear system of equations can be written in matrix form as A x = b , where A is the coefficient matrix, x is the column vector of unknowns, and b is the constant column vector.

1.1 Elimination Method

The elimination method replaces an unknown in one equation with an expression containing another unknown and substitutes it into another equation, thereby eliminating a variable. It is mainly used for solving two‑variable linear systems.

Key properties during elimination:

Swapping two equations does not change the solution.

Multiplying an equation by a non‑zero constant does not change the solution.

Adding a multiple of one equation to another does not change the solution.

1.2 Solutions of a System

A linear system can have three types of solutions:

No solution (inconsistent rows such as 0 = non‑zero ).

Infinitely many solutions (presence of one or more all‑zero rows after elimination).

A unique solution (the system reduces to a perfect echelon form).

1.3 Gaussian Elimination

Gaussian elimination is an algorithm in linear algebra for solving linear systems by performing elementary row operations on the coefficient matrix. It can also be used to determine matrix rank, compute determinants, and find inverses of invertible matrices.

The three elementary row operations are:

Swap two rows.

Multiply a row by a non‑zero scalar.

Add a multiple of one row to another row.

Applying these operations transforms any matrix into an upper‑triangular (row‑echelon) form, and further into a reduced row‑echelon form where each leading entry is 1 and is the only non‑zero entry in its column. The reduced form is unique regardless of the sequence of operations.

2 Augmented Matrix

For a system A x = b , the augmented matrix is formed by appending the constant vector b to the coefficient matrix A as an additional column.

3 Reduced Row‑Echelon Form (RREF)

3.1 Definition

After elementary row transformations, a matrix can be converted to row‑echelon form, characterized by a stair‑step pattern of leading non‑zero entries. When each leading entry is 1 and all other entries in its column are zero, the matrix is in reduced row‑echelon form.

3.2 Procedure

Steps to obtain RREF:

Perform only row operations.

Fix a pivot row (usually the first) and ensure its leading entry is 1.

Use the pivot row to eliminate the corresponding column entries in all other rows.

Proceed to the next column and repeat the process, keeping previously fixed rows unchanged.

3.3 Example

Given points (1,3), (2,‑2), (3,‑5) and (4,0), we seek a cubic polynomial passing through them. Substituting the points into the general cubic p(t)=a t^3 + b t^2 + c t + d yields a system of four equations. Forming the augmented matrix and applying Gaussian elimination reduces it to RREF, from which the coefficients a, b, c, d are obtained, giving the desired polynomial.

4 Summary

This article introduced the concepts of linear systems, elimination methods including Gaussian elimination, the three possible solution cases, and the construction and reduction of augmented matrices to reduced row‑echelon form.