Notes of Linear Algebra for Preparing Mathematical Modeling

Chapter 2: Rectangular Systems and Echlon Forms

Author: Kenneth, S.K. Cheng

Table of Contents

• Notes of Linear Algebra for Preparing Mathematical Modeling
  • Chapter 2: Rectangular Systems and Echlon Forms
    • Author: Kenneth, S.K. Cheng
  • Table of Contents
    • 2.1 Revision of Echlon Form and Rank
    • 2.2 Consistency of Linear Systems

2.1 Revision of Echlon Form and Rank

Recall that a matrix is in echlon form if it is the following form:

$$\begin{bmatrix} 1 & a_{12} & a_{13} & \cdots & a_{1n} \\ 0 & 1 & a_{23} & \cdots & a_{2n} \\ 0 & 0 & 1 & \cdots & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \\ \end{bmatrix}$$

where $a_{ij} = 0$ for $i > j$.

The pivot elements are the first non-zero elements in each row. The pivot columns are the columns containing the pivot elements. The pivot positions are the positions of the pivot elements. We may note that the pivot positions locate on the main diagonal of the matrix; therefore, one may find that Gaussian elimination is a process of transforming a matrix into a triangular matrix.

Definition (Rank): Suppose $A_{m\times n}$ is reduced to echlon form $E_{m\times n}$ by Gaussian elimination. The rank of $A$, denoted by $\operatorname{rank}(A)$, is the number

$$\begin{align*} \operatorname{rank}(A) &= \text{number of pivots} \\ &= \text{number of nonzero rows in } E\\ &= \text{number of basic columns in } A. \end{align*}$$

Theorem 2.1: Let $A$ be a matrix of order $m\times n$. Then $\operatorname{rank}(A) \leq \min(m,n)$.

2.2 Consistency of Linear Systems

A system of linear equations is said to be consistent if it has at least one solution. Otherwise, it is inconsistent.

There are some ways to characterize the consistency(or inconsistency) of a system of linear equations. One of the most common ways is to use the rank of the coefficient matrix.

Suppose there is a system of linear equations $Ax = b$, where $A$ is the coefficient matrix, $x$ is the vector of unknowns, and $b$ is the constant vector. The system is consistent if and only if $\operatorname{rank}(A) = \operatorname{rank}(A|B)$, where $A|B$ is the augmented matrix of the system.

Example: Determine if the following system is consistent:

$$\begin{align*} x_1 + x_2 +2x_3 + 2x_4 + x_5 &= 1\\ 2x_1 + 2x_2 + 4x_3 + 4x_4 + 3x_5 &= 1\\ 2x_1 + 2x_2 + 4x_3 + 4x_4 + 2x_5 &= 2 \\ 3x_1 + 5x_2 + 8x_3 + 6x_4 + 5x_5 &= 3. \end{align*}$$

The augmented matrix is

$$\begin{bmatrix} 1 & 1 & 2 & 2 & 1 & | & 1\\ 2 & 2 & 4 & 4 & 3 & | & 1\\ 2 & 2 & 4 & 4 & 2 & | & 2\\ 3 & 5 & 8 & 6 & 5 & | & 3 \end{bmatrix}.$$

It follows from Gaussian elimination that the original matrix is reduced to

$$\begin{bmatrix} 1 & 1 & 2 & 2 & 1 & | & 1\\ 0 & 0 & 0 & 0 & 1 & | & -1\\ 0 & 0 & 0 & 0 & 0 & | & 0\\ 0 & 2 & 2 & 0 & 2 & | & 0. \end{bmatrix}$$

which is consistent since $\operatorname{rank}(A) = \operatorname{rank}(A|B) = 3$.