Linear systems with two unknowns

Linear systems with two unknowns


	Linear systems with two unknowns

Consider a linear system with two unknowns,

$\displaystyle a_{11} x_1 + a_{12} x_2$	$\textstyle =$	$\displaystyle b_2$
			(1.1)
$\displaystyle a_{21} x_1 + a_{22} x_2$	$\textstyle =$	$\displaystyle b_2$

We can find its solution with the Gaussian elimination procedure. After eliminating

$a_{21}$

$\begin{displaymath} \left( \begin{array}{cc} a_{11} & a_{12} \ a_{21} & a_{22}... ...ft\vert \right. \begin{array}{c} b_1\ b_2 \end{array}\right) \end{displaymath}$

we obtain

$\begin{displaymath} \left( \begin{array}{cc} a_{11} & a_{12} \ 0 & \frac{a_{22... ...b_2\cdot a_{11} - b_2 \cdot a_{21}}{a_{11}} \end{array}\right) \end{displaymath}$

(1.2)

In order to simplify the formulas for the solution of (1.1) we introduce determinant of a matrix,

$\begin{displaymath} \det \left( \begin{array}{cc} a_{11} & a_{12} \ a_{21} & a... ...\end{array} \right) = a_{11}\cdot a_{22} - a_{12} \cdot a_{21} \end{displaymath}$

It is convenient to write system (1.1) as

$\begin{displaymath} a_1 x_1 + a_2 x_2 = b, \end{displaymath}$

where

$\begin{displaymath} a_1 = \left( \begin{array}{c} a_{11}\ a_{21} \end{array... ...left( \begin{array}{c} b_{1}\ b_{2} \end{array} \right) \end{displaymath}$

In accordance with this notations

$\begin{displaymath} \det \left( \begin{array}{cc} a_{11} & a_{12} \ a_{21} & a... ...} \end{array} \right) = \det \left( a_{1} \;\; a_{2} \right) \end{displaymath}$

and it follows from (1.2) that if

$\begin{displaymath} det (a_1 \; a_2 ) \not= 0 \end{displaymath}$

then there exists the only solution for (1.1) given by

$\displaystyle x_1$	$\textstyle =$	$\displaystyle \frac{\det (b \; a_2 )}{det (a_1 \; a_2 )}$
			(1.3)
$\displaystyle x_2$	$\textstyle =$	$\displaystyle \frac{\det (a_1 \; b )}{det (a_1 \; a_2 )}$

These formulas are known as Cramer's rule (about Cramer see http://www-groups.dcs.st-and.ac.uk/history/Mathematicians/Cramer.html).

They are remarkable because they admit a straightforward generalization to any linear system having n unknowns and n equations. Indeed, consider

$\begin{displaymath} a_1 x_1 + a_2 x_2 +\dots + a_n x_n = b, \end{displaymath}$

(1.4)

where

$\begin{displaymath} a_i=\left( \begin{array}{c} a_{1i} \ a_{2i} \ \vdots ... ...ay}{c} b_1 \ b_2 \ \vdots \ b_n \end{array} \right) \end{displaymath}$

Then Cramer's rule gives us the only solution for (1.4) as long as

$\begin{displaymath} det(a_1\;a_2 \dots a_n) \not= 0 \end{displaymath}$

and the solution is defined as follows.

$\begin{eqnarray*} x_1 &=& \frac{\det(b \; a_2 \; \dots a_n)}{\det(a_1 \; a_2 \; ... ...frac{\det(a_1 \; a_2 \; \dots b)}{\det(a_1 \; a_2 \; \dots a_n)} \end{eqnarray*}$

Next: Determinant Up: Cramer's rule Previous: Cramer's rule

Sergey Nikitin 2004-09-09