Question

Define row-reduced echeion form and give examples.

Short answer

The row-reduced echelon form (RREF) is a matrix that has been transformed using elementary row operations to satisfy the following properties: 1) it is in echelon form, 2) every pivot is equal to 1, and 3) each pivot is the only nonzero entry in its column. Examples of RREF include: Example 1: \[ \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \] Example 2: \[ \begin{bmatrix} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & 4 \end{bmatrix} \]

Step-by-step solution

Definition of Echelon Form

An echelon form (also called row echelon form or REF) of a matrix is a matrix that has been transformed using elementary row operations such that: 1. All nonzero rows are above any rows of all zeros. 2. The leading entry (also called pivot) of a nonzero row is to the right of the pivot of the row above it. 3. All entries below a pivot are zeros.

Definition of Row-Reduced Echelon Form

The row-reduced echelon form (RREF) is an echelon form of a matrix with additional properties: 1. Every pivot is equal to 1. 2. Each pivot is the only nonzero entry in its column.

Provide Examples of Echelon Form

Here are two examples of matrices in echelon form: Example 1: \[ \begin{bmatrix} 1 & 2 & 4 \\ 0 & 3 & 6 \\ 0 & 0 & 5 \end{bmatrix} \] Example 2: \[ \begin{bmatrix} 1 & 0 & 2 & 3 \\ 0 & 1 & 4 & 5 \\ 0 & 0 & 0 & 6 \end{bmatrix} \] Notice that both matrices satisfy the three properties of the echelon form.

Provide Examples of Row-Reduced Echelon Form

Here are two examples of matrices in row-reduced echelon form: Example 1: \[ \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \] Example 2: \[ \begin{bmatrix} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & 4 \end{bmatrix} \] Notice that both matrices satisfy the properties of the echelon form and also the additional properties of the row-reduced echelon form.

Key concepts

Echelon Form

The concept of echelon form, also known as row echelon form (REF), is fundamental in matrix operations. It's a stage in a matrix where the structure follows a specific order, aiding in solving linear equations. The structure is:

• All non-zero rows appear above any rows of zeros.
• Each leading entry, or pivot, in a non-zero row is strictly to the right of the leading entry of the row above it.
• All entries below the pivots are zeros.

Echelon forms are like stepping stones to simplification. They make it easier to analyze matrices or solve linear systems by exploring relationships between variables.

Matrix Operations

Matrix operations are procedures that encompass addition, subtraction, multiplication, and changes applied to matrices. When transforming a matrix to its echelon form, we often use operations such as:

• Row Interchange - swapping two rows of the matrix.
• Row Scaling - multiplying a row by a non-zero constant.
• Row Addition - adding a multiple of one row to another row.

These operations help pave the way to refine the matrix into simpler forms, like the echelon forms, facilitating easier computations and interpretations.

Pivot Elements

Pivot elements are the foundations or markers in a matrix transformation process. Think of them as the key holders unlocking the optimal reduction of a matrix:

• Pivots are the first non-zero numbers in each non-zero row.
• They are placed in increasing column order from top to bottom, dictating the structure of the echelon form.
• In row-reduced echelon form, pivots are also the only non-zero entries in their columns and are normalized to 1.

Identifying and manipulating pivot elements is crucial for achieving forms that allow for straightforward solutions of linear equations.

Elementary Row Operations

Elementary row operations are the authorized moves or manipulations allowed on matrices to achieve specific forms such as echelon form or row-reduced echelon form. These operations must follow given rules, consisting of:

• Swapping rows - changing the positions of two rows.
• Multiplying a row by a non-zero scalar - adjusting row magnitudes without altering their zeros and signs.
• Adding or subtracting a multiple of one row to another – this creates zeros below or above pivot elements, vital for reaching echelon form.

Performing these operations methodically leads to cleaner matrix forms, enabling better interpretations and simplifications within mathematical contexts.