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Section 15.2 Matrices and Systems of Linear Equations

Definition 15.2.1.

An \(m\times n\) matrix is a rectangular array of numbers with \(m\) rows and \(n\) columns, i.e.

\begin{equation*} \begin{pmatrix} 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{pmatrix}=(a_{ij})_{m\times n} \end{equation*}

We say that the number \(a_{ij}\) is the \((i,j)\)th entry (i.e. the entry in row \(i\text{,}\) column \(j\)).

The following are examples of matrices.

\begin{equation*} \begin{pmatrix} 1 \amp 2 \amp 3 \\ 4 \amp 5 \amp 10 \end{pmatrix} \hspace{5mm} \begin{pmatrix} 0 \amp \frac{1}{4} \\ -\frac{1}{2} \amp 3 \end{pmatrix} \hspace{5mm} \begin{pmatrix} \sqrt{2} \\ 1 \\ -3 \end{pmatrix} \end{equation*}
\begin{equation*} \hspace{5mm} 2\times 3 \textrm{ matrix } \hspace{5mm} 2\times 2 \textrm{ matrix } \hspace{4mm} 3\times 1 \textrm{ matrix } \end{equation*}

Matrices have proved useful in many areas of mathematics and we will study them in more detail throughout this strand. With respect to the following system of linear equations

\begin{align*} a_{11}x_1+a_{12}x_2+\dots+a_{1n}x_n \amp =b_1\\ a_{21}x_1+a_{22}x_2+\dots+a_{2n}x_n \amp =b_2\\ \vdots \\ a_{m1}x_1+a_{m2}x_2+\dots+a_{mn}x_n \amp =b_m \end{align*}

so long as the variables are always written in the same order and the constants are always put on to the right hand side we can represent this system via the \(m\times (n+1)\) matrix

\begin{equation*} \begin{pmatrix} 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_m \end{pmatrix} \end{equation*}

This matrix is called the augmented matrix for the system of linear equations. Each row of this matrix corresponds to an equation in the system. Each of the first \(n\) columns corresponds to the coefficients of a variable in the equations and the last column gives the constants on the right hand sides of the equations.

Write down the augmented matrix for the following non-homogenous system of linear equations

\begin{align*} 3x_1-x_2+2x_3+x_4 \amp =2\\ -x_1+6x_2-11x_4 \amp =-22\\ x_1+x_2+7x_3+4x_4 \amp =8 \end{align*}
Solution.

The augmented matrix will be the \(3\times 5\) matrix

\begin{equation*} \begin{pmatrix} 3 \amp -1 \amp 2 \amp 1 \amp 2 \\ -1 \amp 6 \amp 0 \amp -11 \amp -22 \\ 1 \amp 1 \amp 7 \amp 4 \amp 8 \end{pmatrix} \end{equation*}

Note that sometimes a partition line is put before the last column in an augmented matrix to emphasise that the last column represents the right hand side of the equations while the remaining columns represent the coefficients of the variables in the system. The matrix without the last column is called the coefficient matrix for the system of equations.

For this system of linear equations the augmented matrix is sometimes written as

\begin{equation*} \left(\begin{array}{c c c c | c} 3 \amp -1 \amp 2 \amp 1 \amp 2 \\ -1 \amp 6 \amp 0 \amp -11\amp -22 \\ 1 \amp 1 \amp 7 \amp 4 \amp 8 \end{array}\right) \end{equation*}

The coefficient matrix for this system is

\begin{equation*} \begin{pmatrix} 3 \amp -1 \amp 2 \amp 1 \\ -1 \amp 6 \amp 0 \amp -11 \\ 1 \amp 1 \amp 7 \amp 4 \end{pmatrix} \end{equation*}

Write down the system of linear equations that correspond to the following augmented matrices.

  1. \begin{equation*} \left(\begin{array}{c c c | c} 3 \amp -2 \amp 2 \amp -3 \\ -1 \amp 6 \amp 4 \amp 7 \\ 1 \amp 1 \amp 5 \amp -2 \end{array}\right) \end{equation*}
  2. \begin{equation*} \left(\begin{array}{c c c | c} 1 \amp 0 \amp 0 \amp 1 \\ 0 \amp 1 \amp 0 \amp 2 \\ 0 \amp 0 \amp 1 \amp -1 \end{array}\right) \end{equation*}
Solution.

  1. Calling the variables as \(x\text{,}\) \(y\) and \(z\) the system of equations is

    \begin{align*} 3x-2y+2z \amp =-3\\ -x+6y+4z \amp =7\\ x+y+5z \amp =-2 \end{align*}

  2. Again calling the variables \(x\text{,}\) \(y\) and \(z\) this augmented matrix represents the very simple system of equations

    \begin{align*} x \amp =1\\ y \amp =2\\ z \amp =-1 \end{align*}