title: Systems: Matrices¶

Matrices¶

It is time to use some better tools to do the bookkeeping needed to solve linear systems. Matrices help doing that. Later on (§Chapter 4), we will use matrices in a much more profound way.

Definition 2.20

A matrix is a rectangular array of numbers. We speak of an \(m \times n\)-matrix (or \(m\)-by-\(n\) matrix) if it has \(m\) rows and \(n\) columns, respectively. If \(m=n\), we also call it a square matrix.

An \(1 \times n\)-matrix (i.e., \(m=1\) and \(n\) is arbitrary) is called a row vector. Similarly, an \(m \times 1\)-matrix is called a column vector.

Example 2.21

It is customary to denote matrices by capital letters. For example,

\[ A = \left ( \begin{array}{cc} 3 & 4 \\ 0 & -7 \end{array} \right ) \]

is a \(2 \times 2\)-matrix (or square matrix of size 2).

\[ B = \left ( \begin{array}{ccc} 1 & -2 & 0 \\ 1 & 0 & 3 \end{array} \right ) \]

is a \(2 \times 3\)-matrix and

\[ C = \left ( \begin{array}{cc} 1 & -2 \\ 0 & 1 \\ 0 & -3 \end{array} \right ) \]

is a \(3 \times 2\)-matrix.

The entries of a matrix may also be variables. For example

\[ \left ( \begin{array}{c} x_1 \\ x_2 \end{array} \right ) \]

is a column vector (or a \(2 \times 1\)-matrix), whose entries are two variables; \(\left ( \begin{array}{cc} x_1 & x_2 \end{array} \right )\) is a row vector (or a \(1 \times 2\)-matrix).

Notation 2.22

A matrix whose entries are unspecified numbers is denoted like so:

\[ A = \left ( \begin{array}{ccccc} a_{11} & a_{12} & a_{13} & \dots & a_{1n} \\ a_{21} & a_{22} & a_{23} & \dots & a_{2n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & a_{m3} & \dots & a_{mn}\\ \end{array} \right ). \]

Thus, the number \(a_{ij}\) is the entry in the \(i\)-th row and the \(j\)-th column. A more compressed notation expressing the same is

\[ A = (a_{ij})_{i=1, \dots, m, j = 1, \dots, n} \]

or even just

\[ A = (a_{ij}). \]

Definition 2.23 (Related exercises: Exercise 4.46)

Let

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

(2.24)

be a linear system (consisting of \(m\) equations, in the unknowns \(x_1, \dots, x_n\); the numbers \(a_{ij}\) are the coefficients, the numbers \(b_1, \dots, b_n\) are the constants).

The matrix associated to this system is the following \(m \times (n+1)\)-matrix (the vertical bar is just there to remind ourselves that the last column corresponds to the constants in the equations above; one also speaks of an augmented matrix)

\[ A = \left ( \begin{array}{cccc|c} a_{11} & a_{12} & \dots & a_{1n} & b_1 \\ a_{21} & a_{22} & \dots & a_{2n} & b_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{m1} & a_{m2} & \dots & a_{mn} & b_m \\ \end{array} \right ). \]

(2.25)

In other words, the matrix is the rectangular array containing the coefficients and the constants of the individual equations, and suppresses the mention of the variables.

Example 2.26

The matrix associated to the system

\[ \begin{align*} x+y&=4 \\ x-y&=1 \end{align*} \]

is the \(2 \times 3\)-matrix

\[ \left ( \begin{array}{cc|c} 1 & 1 & 4 \\ 1 & -1 & 1 \end{array} \right ). \]

Of course, the process of associating a matrix to a linear system can be reversed since any \(m \times (n+1)\)-matrix gives rise to a linear system: the matrix gives rise to the linear system . For example, the \(2 \times 3\)- matrix

\[ \left ( \begin{array}{cc|c} 1 & -2 & 0 \\ 1 & 0 & 3 \end{array} \right ) \]

gives rise to the linear sytem

\[ \begin{align*} x - 2y & = 0 \\ x & = 3. \end{align*} \]