Matrices and Vectors

When a system of equations has more than two or three equations, it is difficult to discuss them without using matrices and vectors.
A matrix is a rectangular array of numbers in which not only the value of the number is important but also its position in the array.

$\begin{indisplay} A=\left[ \begin{array}{cccc} a_{11} & a_{12} & \ldots & a_{1m... ...in{array}{c} i=1,2,\ldots,n, \\ j=1,2,\ldots,m \\ \end{array}\end{indisplay}$
An $m \times n$ matrix times as $n \times 1$ vector gives an $m \times 1$ product ( $m \times \textbf{n} \textbf{n} \times 1$ ) .
The general relation for $\boxed{\textbf{$Ax=b$}}$ is

$\displaystyle b_i=\sum_{k=1}^{No.of.cols.}a_{ik}x_{k},~~~~~~~i=1,2,\ldots,\char93 ~ of~rows$
where is a matrix, and are vectors (column vectors).

Much more simply in matrix notation, as where

$\begin{indisplay} A=\left[ \begin{array}{cccc} a_{11} & a_{12} & \ldots & a_{1n... ...rray}{c} b_1\\ b_2\\ \vdots \\ b_n\\ \end{array}\right] \end{indisplay}$
Example,

Matrix notation:

$\begin{indisplay} \left[ \begin{array}{rrr} 3 & 2 & 4 \\ 1 & -2 & 0 \\ -1 &... ...=\left[ \begin{array}{r} 14\\ -7\\ 2\\ \end{array}\right] \end{indisplay}$

$\displaystyle \Leftrightarrow$

Set of equations:

$\displaystyle 3x_1 + 2x_2 + 4x_3$ $\displaystyle =$ $\displaystyle 14$

$\displaystyle x_1 - 2x_2$ $\displaystyle =$ $\displaystyle -7$

$\displaystyle -x_1 + 3x_2 + 2x_3$ $\displaystyle =$ $\displaystyle 2$
Square matrices are particularly important when a system of equations is to be solved.