What are free variables in linear systems?

Free variables are variables that correspond to columns without pivots in the RREF of a coefficient matrix. Free variables can take on any arbitrary value, which leads to infinitely many solutions in a consistent system.

How do you solve Ax = B with RREF?

Write the system as [A | B], reduce to RREF. If consistent, read the solutions directly: pivot columns give values of pivot variables in terms of constants and free variables.

What are elementary row operations?

Elementary row operations are three types of operations that can be performed on a matrix without changing the solution set of the system it represents: 1. Swapping two rows. 2. Scaling a row by a non-zero constant. 3. Adding or subtracting a multiple of one row to another.

Why is exact fraction arithmetic important in matrices?

Using floating-point numbers in computers can lead to rounding errors, turning exact zeros into values like 1e-16. This breaks pivot detection and yields inaccurate RREF steps. Exact fraction math prevents this.

What is an Augmented Matrix? Setup and Reduction in Linear Algebra

Definition of an Augmented Matrix

An augmented matrix is created by appending the columns of one matrix onto another matrix. The vertical divider bar represents the equal sign in systems of equations or separates different matrix zones during calculations. The bar is purely visual and does not affect row operations.

The two most common applications of augmented matrices are:

Representing systems of linear equations: [A | B], where A is the coefficient matrix and B is the constant vector.
Finding matrix inverses: [A | I], where A is the square matrix to invert and I is the Identity matrix.

Representing Systems of Equations

Consider a system of three linear equations with three variables:

1x₁ + 2x₂ + 3x₃ = 9
2x₁ - 1x₂ + 1x₃ = 8
3x₁ + 0x₂ - 1x₃ = 3

We extract the coefficients of the variables to form matrix A, and the constants to form vector B. The augmented matrix [A | B] is written as:

1	2	3	9
2	-1	1	8
3	0	-1	3

Why We Perform Row Operations on the Augmented Matrix

Elementary row operations represent algebraic manipulations of the original system of equations: swapping two equations, multiplying an equation by a scalar, or adding a multiple of one equation to another. To maintain algebraic equivalence, any operation performed on the left-hand coefficients (matrix A) must also be performed on the right-hand constants (vector B). The augmented matrix ensures that we do not forget to update the constants during row reduction.

Interpreting the Reduced Augmented Column

After reducing the augmented matrix to RREF, we can immediately read the status of the system:

Unique Solution: If the coefficients side reduces to the Identity matrix, the augmented column contains the unique solution values: [1 0 | x], [0 1 | y], meaning x = 5, y = -2.
Infinite Solutions: If a column does not have a pivot, its variable is a free variable. This indicates the system has infinitely many solutions, which can be parameterized.
No Solution (Inconsistent): If row reduction yields a row like [0, 0, 0 | 1] (where coefficients are all zero, but the constant is non-zero), the system represents the equation 0 = 1, which is mathematically impossible. Therefore, the system is inconsistent and has no solution.

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