Linear Equations

MATLAB

Consider m linear equations in n unknowns namely of x_1 ,x_2 , . . . , x_n  as:

a_{11}x_1 + a_{12}x_2 + ... + a_{1n}x_n = b_{1} \\ a_{21}x_1 + a_{22}x_2 + ... + a_{2n}x_n = b_{2} \\ . \\ . \\ . \\ a_{m1}x_1 + a_{m2}x_2 + ... + a_{mn}x_n = b_{m}

Equivalently,

Ax =b

Associated with this system of equations is the matrix:

A =[a_1 , a_2 , . . . , a_n]

Consider the m\times n matrix

\[ \left( \begin{array}{cccc} a_{11} & a_{21} & ... & a_{m1} \\ a_{12} & a_{22} & ... & a_{m2} \\ . & . & . & . \\ . & . & . & . \\ a_{1n} & a_{2n} & ... & a_{mn} \\ \end{array} \right)\]

We can apply elementary row operations in the matrix A to get the matrix in reduced form.

An elementary row operation on the given matrix A is an algebraic manipulation of the matrix that corresponds to one of the following:

1. Interchanging any two rows such as the p^{th} and the u^{th} rows of the matrix A;

2. Multiplying one of its rows such as the p^{th} row by a real number \alpha where \alpha \neq 0 ;

3. Adding one of its rows such as the u^{th} row to the \beta times p^{th} row.

 

 

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