### Systems of linear equations and matrices: Systems and matrices

### From systems to matrices

The *elimination method*, which solves systems of linear equations by use of elementary operations, actually works only with the coefficients and constants of the system. A good accounting in the form of a succinct notation can help expedite this process.

The system of \(m\) linear equations with \(n\) unknowns \(x_1, \ldots, x_n\) \[\left\{\;\begin{array}{llllllllll} a_{11}x_1 \!\!\!\!&+&\!\!\!\! a_{12}x_2 \!\!\!\!&+&\!\!\!\! \cdots \!\!\!\!&+&\!\!\!\! a_{1n}x_n\!\!\!\!&=&\!\!\!\!b_1\\ a_{21}x_1 \!\!\!\!&+&\!\!\!\! a_{22}x_2 \!\!\!\!&+&\!\!\!\! \cdots \!\!\!\!&+&\!\!\!\! a_{2n}x_n\!\!\!\!&=&\!\!\!\!b_2\\ \;\;\vdots && \vdots &&&& \vdots&&\!\!\!\!\vdots\\ a_{m1}x_1 \!\!\!\!&+&\!\!\!\! a_{m2}x_2 \!\!\!\!&+&\!\!\!\! \cdots \!\!\!\!&+&\!\!\!\! a_{mn}x_n\!\!\!\!&=&\!\!\!\!b_m\end{array}\right.\] is often written as follows:

\[\begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & & \vdots \\ a_{m1} & \cdots & a_{mn} \end{pmatrix}\!\!\begin{pmatrix}x_1\\ \vdots \\ x_n\end{pmatrix}= \begin{pmatrix}b_1\\ \vdots \\ b_m\end{pmatrix}\]

Such a rectangular array is called a **matrix**, and is often framed in round brackets. Since the unknowns \(x_1, \ldots, x_n\) and their order of appearance does not change during the solving process, the system is also well represented by the matrix \[\left(\,\begin{array}{cccc|c}

a_{11} & a_{12} & \cdots & a_{1n} & b_1\\ a_{21} & a_{22} & \cdots & a_{2n} & b_2\\ \vdots & \vdots & & \vdots & \vdots\\ a_{m1} & a_{m2} & \cdots & a_{mn} & b_m\end{array}\,\right)\] The vertical line is drawn to indicate that the last column represents the right-hand side of the system. In general, however, this line is omitted. This matrix is called the **augmented matrix**; "augmented" because the matrix can be viewed as \[(A|\vec{b})\] where \(A\) stands for the **coefficient matrix** of the system and \(\vec{b}\) for the column of numbers at the right-hand sides of the equations: \[A=\begin{pmatrix}a_{11} & a_{12} & \cdots & a_{1n} \\

\vdots & \vdots & & \vdots \\

a_{m1} & a_{m2} & \cdots & a_{mn}\end{pmatrix}\quad\mathrm{and}\quad \vec{b}=\cv{b_1 \cr \vdots\cr b_m\cr}\]

With this notation and with # \vec {x} = \begin{pmatrix}x_1\\ \vdots \\ x_n\end{pmatrix} #, we can briefly describe the above system as \[A\vec{x} = \vec{b}\]

It is also possible to click on the following expression, and to replace the three little boxes by the appropriate equations:

\[\lineqs{&\Box& \\ &\Box& \\ &\Box & \\ }\]

Do not carry out any simplifications, but make the direct translation from the matrix to the system.

This system of linear equations entered in at least two ways:

- by using the logic and, #\land#, which can be entered by clicking its icon on the virtual palette under the tab logic.
- by clicking on the left brace on the virtual palette under the tab vector; it is set for two rows, but you can (in due course *** ***) you can add a row by pressing the down arrow on your keyboard.

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