Introduction to Matrices, Augmented Matrix of a System

Section 2.3 Introduction to Matrices, Augmented Matrix of a System

Notice that a system of linear equations can be essentially represented by its coefficients. Given a system,

\begin{alignat*}{2} x \amp + 2y \amp = 5\\ 2x \amp + 3y \amp = 8 \end{alignat*}

it can be represented fully by the following array of coefficients,

\begin{equation*} \begin{bmatrix} 1 \amp 2 \amp 5 \\ 2 \amp 3 \amp 8 \end{bmatrix} \end{equation*}

This array captures the essential information of the system. That is, the variables in the original system simply act as “placeholders”. As long as we keep track of which columns correspond to which variables, the array contains precisely the same information about the system as its “equation form”. Systems of linear equations can be conveniently and efficiently represented in this way. This kind of rectangular array of numbers is called a matrix.

Subsection 2.3.1 Matrices

Broadly, a matrix is a rectangular array of numbers, arranged in (horizontal) rows and (vertical) columns, and within brackets.

Definition 2.3.1.

Let \(m, n \in \mathbb{N}\text{.}\) Then, an \(m \times n\) matrix (read as “\(m\) by \(n\) matrix”) is a rectangular array of \(m\) rows and \(n\) columns of real numbers,

\begin{equation*} \begin{bmatrix} a_{11} \amp a_{12} \amp \dots \amp a_{1n} \\ a_{21} \amp a_{22} \amp \dots \amp a_{2n} \\ \vdots \amp \vdots \amp \ddots \amp \vdots \\ a_{m1} \amp a_{m2} \amp \dots \amp a_{mn} \end{bmatrix} \end{equation*}

Each number \(a_{ij}\) is called an entry (or coefficient, or element) of the matrix. Then, \(a_{ij}\) represents the entry in row \(i\) and column \(j\text{.}\)

The plural of matrix is matrices
Matrices are typically denoted by capital letters, like \(A, B, C\text{,}\) etc.

Note that in \(m \times n\text{,}\) the number of rows is specified first, then the number of columns.

Definition 2.3.2.

A matrix \(A\) is square if it has the same number of rows as columns. In other words, \(A\) is an \(n \times n\) matrix, for some \(n\text{.}\)

Examples will often include \(2 \times 2\) or \(3 \times 3\) matrices. A general \(2 \times 2\) matrix is of the form,

\begin{equation*} A = \begin{bmatrix} a_{11} \amp a_{12} \\ a_{21} \amp a_{22} \end{bmatrix} \qquad \text{or} \qquad A = \begin{bmatrix} a \amp b \\ c \amp d \end{bmatrix} \end{equation*}

The latter notation is simpler because it doesn't involve subscripts. However, if there are multiple matrices involved, the former notation can be more practical.

Subsection 2.3.2 Augmented Matrix of a System

Consider the linear system,

\begin{equation*} \begin{array}{rrrrr} x_1 \amp - 3x_2 \amp + 4x_3 \amp = \amp -4 \\ 3x_1 \amp \amp + 7x_3 \amp = \amp -8 \\ -4x_1 \amp + 6x_2 \amp - x_3 \amp = \amp 7 \end{array} \end{equation*}

\(3 \times 3\)\(3 \times 1\)

\begin{equation*} \begin{bmatrix} 1 \amp -3 \amp 4 \\ 3 \amp 0 \amp 7 \\ -4 \amp 6 \amp -1 \end{bmatrix} \qquad \begin{bmatrix} -4 \\ -8 \\ 7 \end{bmatrix} \end{equation*}

\(3 \times 4\)

\begin{equation*} \begin{bmatrix} 1 \amp -3 \amp 4 \amp -4 \\ 3 \amp 0 \amp 7 \amp -8 \\ -4 \amp 6 \amp -1 \amp 7 \end{bmatrix} \end{equation*}

augmented matrix\(x_1, x_2, x_3\)

\begin{equation*} \left[\begin{array}{ccc|c} 1 \amp -3 \amp 4 \amp -4 \\ 3 \amp 0 \amp 7 \amp -8 \\ -4 \amp 6 \amp -1 \amp 7 \end{array}\right] \end{equation*}

Definition 2.3.3.

The coefficient matrix of a linear system is the matrix with entries given by the coefficients of the variables. The augmented matrix is the coefficient matrix with an additional column for the constant terms. The system,

\begin{alignat*}{3} a_{11}x_1 \amp + a_{12}x_2 \amp + \dots \amp + a_{1n}x_n \amp = b_1\\ a_{21}x_1 \amp + a_{22}x_2 \amp + \dots \amp + a_{2n}x_n \amp = b_2\\ \vdots \quad \amp \qquad \vdots \amp \amp \qquad \vdots \amp \vdots\\ a_{m1}x_1 \amp + a_{m2}x_2 \amp + \dots \amp + a_{mn}x_n \amp = b_n \end{alignat*}

has coefficient matrix and augmented matrix,

\begin{equation*} \begin{array}{cc} \underbrace{\begin{bmatrix} a_{11} \amp a_{12} \amp \dots \amp a_{1n} \\ a_{21} \amp a_{22} \amp \dots \amp a_{2n} \\ \vdots \amp \vdots \amp \ddots \amp \vdots \\ a_{m1} \amp a_{m2} \amp \dots \amp a_{mn} \end{bmatrix}}_{\text{coefficient matrix}} \amp \underbrace{\left[\begin{array}{cccc|c} a_{11} \amp a_{12} \amp \dots \amp a_{1n} \amp b_1 \\ a_{21} \amp a_{22} \amp \dots \amp a_{2n} \amp b_2 \\ \vdots \amp \vdots \amp \ddots \amp \vdots \amp \vdots \\ a_{m1} \amp a_{m2} \amp \dots \amp a_{mn} \amp b_n \end{array}\right]}_{\text{augmented matrix}} \end{array} \end{equation*}

Quite literally, the augmented matrix results from “augmenting” the constant terms onto the coefficient matrix.

Linear Algebra Notes