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

### Equations and matrices

Now that we have discussed how systems of linear equations can be represented by means of matrices, we formulate the *elementary row operations on systems of linear equations* in terms of matrices.

Elementary row operations The elementary operations on systems of linear equations translate directly into **elementary row operations** on (or **elementary operations on rows** of) the augmented matrix. These operations can be briefly described by denoting the \(i\)-th row of the matrix by \(R_i\). The three elementary row operations on matrices are:

- \(R_i\rightarrow c\cdot R_i\;(c\neq 0)\): multiplying all the numbers in the \(i\)-th row by the number \(c\). \[ \small\left(\begin{array}{ccc|c} a_{11} & \cdots & a_{1n} & b_1\\ \vdots && \vdots & \vdots \\ a_{i1} & \cdots & a_{in} & b_i \\ \vdots && \vdots & \vdots \\ a_{m1} & \cdots & a_{mn} & b_m\end{array}\right)\rightsquigarrow \left(\begin{array}{ccc|c} a_{11} & \cdots & a_{1n} & b_1\\ \vdots && \vdots & \vdots \\ c\cdot a_{i1} & \cdots & c\cdot a_{in} & c\cdot b_i \\ \vdots && \vdots & \vdots \\ a_{m1} & \cdots & a_{mn} & b_m\end{array}\right)\]
- \(R_i\rightarrow R_i+c\cdot R_j\): adding the multiple by \(c\) of the \(j\)-th row to the \(i\)-th row. \[\small \left(\begin{array}{ccc|c} a_{11} & \cdots & a_{1n} & b_1\\ \vdots && \vdots & \vdots \\ a_{i1} & \cdots & a_{in} & b_i \\ \vdots && \vdots & \vdots \\ a_{j1} & \cdots & a_{jn} & b_j \\ \vdots && \vdots & \vdots\\ a_{m1} & \cdots & a_{mn} & b_m\end{array}\right)\rightsquigarrow \left(\begin{array}{ccc|c} a_{11} & \cdots & a_{1n} & b_1\\ \vdots && \vdots & \vdots \\ a_{i1}+c\, a_{j1} & \cdots & a_{in}+c\, a_{jn} & b_{i}+c\, b_j \\ \vdots&& \vdots & \vdots \\ a_{j1}& \cdots & a_{jn} & b_j \\ \vdots && \vdots & \vdots\\ a_{m1} & \cdots & a_{mn} & b_m\end{array}\right)\]
- \(R_i\leftrightarrow R_j\): interchanging the \(i\)-th row and \(j\)-th row. \[\small\left(\begin{array}{ccc|c} a_{11} & \cdots & a_{1n} & b_1\\ \vdots && \vdots & \vdots \\ a_{i1} & \cdots & a_{in} & b_i \\ \vdots && \vdots & \vdots \\ a_{j1} & \cdots & a_{jn} & b_j \\ \vdots && \vdots & \vdots\\ a_{m1} & \cdots & a_{mn} & b_m\end{array}\right)\rightsquigarrow \left(\begin{array}{ccc|c} a_{11} & \cdots & a_{1n} & b_1\\ \vdots && \vdots & \vdots \\ a_{j1} & \cdots & a_{jn} & b_j \\ \vdots&& \vdots & \vdots \\ a_{i1}& \cdots & a_{in} & b_i \\ \vdots && \vdots & \vdots\\ a_{m1} & \cdots & a_{mn} & b_m\end{array}\right)\]

The arrow \(\rightarrow\) in the above row operations has the meaning of "is replaced by". This describes the row operation(s) we carried out to transform the matrix on the left of the symbol into the matrix on the right.

The symbol \(\rightsquigarrow\) has the meaning "is reduced to". We use the symbol \(\boldsymbol{\sim}\) to indicate that the matrices on the left and on the right of this symbol arose from one another by a series of elementary row operations. We then say that one matrix is obtained from the other by **row reduction **or briefly** reduction**.

A matrix obtained by reduction from the augmented matrix of a system of linear equations is the augmented matrix of a system of equations having the same solution as the original system.

We will address the translation of the solving of a system of linear equations in terms of matrices. Here is the first step in this direction:

The goal

Each elementary operation of a matrix corresponds to an elementary operation on the associated system of linear equations. Therefore, systems of equations associated with matrices #A# and #B# such that \(A \rightsquigarrow B\) are equivalent.

The intention is to alter the augmented matrix by use of elementary operations in such a way that the associated system of equations will be solved.

For example, if the left-hand part of the coefficient matrix consisting of the elements from all #m# rows in the first #m# columns, has the form \[\matrix{1&0&0&\cdots&0 \\ 0&1&0&\cdots&0 \\ \vdots&\vdots&&\vdots&0 \\ 0&\cdots&\cdots&1&0 \\ 0&\cdots&\cdots&0&1 }\] then the solution of the system of equations can be read off from the augmented matrix: the unknowns #x_1,\ldots,x_m# are expressed in terms of the constants and the remaining unknowns #x_{m+1},\ldots,x_n# (which are begin used here as free parameters). The above matrix is called the **identity matrix** of dimension #m# and is denoted by #I_m# or even #I# if the dimension is clear.

You are encouraged to consider some examples of systems with two or three unknowns for a taste of the elimination method by use of matrices.

In order to see this, we begin with the original system of equations \[\lineqs{ 3 x-3 y&=&-3\cr 3 x&=&-1 \cr}\] and write it down the corresponding augmented matrix. We will then perform elementary operations and continue the reduction process until we have a specific form that will be described in general in later theory, but that implies here that the coefficient matrix is the identity matrix \(\matrix{1&0\\ 0&1}\). Then the solution of the system of equations can be read off easily from the newly obtained augmented matrix. The matrix elements that have the desired value during the following reduction process are coloured green. We can work with different elementary operations in intermediate steps, but we will always arrive at the same result.

\[\begin{array}{rcll} \left(\begin{array}{rr|r}3 & -3 & -3 \\ 3 & 0 & -1 \end{array}\right) & \sim & \left(\begin{array}{rr|r} 3 & -3 & -3 \\ \color{green}{0} & 3 & 2 \end{array}\right) &\color{blue}{\begin{array}{l} \phantom{x} \\ R_{2}-R_{1}\end{array}}\\ \\ & \sim & \left(\begin{array}{rr|r} 3 & \color{green}{0} & -1 \\ \color{green}{0} & 3 & 2 \end{array}\right) &\color{blue}{\begin{array}{l} R_{2}+R_{1}\\ \phantom{x}\end{array}} \\ \\ & \sim & \left(\,\begin{array}{rr|r} \color{green}{1}& \color{green}{0} & -{{1}\over{3}} \\ \color{green}{0} & \color{green}{1} & {{2}\over{3}}\end{array}\,\right) &\color{blue}{\begin{array}{r}\frac{1}{3}\!R_1\\ \frac{1}{3}\!R_2\end{array}} \end{array}\] The solution of the system of equations is now easy to read off: \[x=-{{1}\over{3}}\land y={{2}\over{3}}\]

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