PlanetMath: example of Cramer's rule

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example of Cramer's rule

(Example)

Say we want to solve the system

$\displaystyle 3x + 2y + z -2w$	$\displaystyle =$	$\displaystyle 4$
$\displaystyle 2x - y + 2z - 5w$	$\displaystyle =$	$\displaystyle 15$
$\displaystyle 4x + 2y \phantom{+ 2x} - 5w$	$\displaystyle =$	$\displaystyle 1$
$\displaystyle 3x \phantom{+ 2y} - 2z -4w$	$\displaystyle =$	$\displaystyle 1.$

The associated matrix is

$\displaystyle \begin{pmatrix} 3 & 2 & 1 & -2\ 2 & -1 & 2 & -5\ 4 & 2 & 0 & -1 \ 3 & 0 & -2 & -4 \end{pmatrix}$

whose determinant is $\Delta=-65$ . Since the determinant is non-zero, we can use Cramer's rule. To obtain the value of the -th variable, we replace the -th column of the matrix above by the column vector

$\displaystyle \begin{pmatrix}4\\ 15\\ 1\\ 1\end{pmatrix},$

the determinant of the obtained matrix is divided by $\Delta$ and the resulting value is the wanted solution.

$\displaystyle x=\frac{\Delta_1}{\Delta}= \frac{ \begin{vmatrix} 4 & 2 & 1 & -2\... ... -5\ 1 & 2 & 0 & -1\ 1 & 0 & -2 & -4 \end{vmatrix}}{-65}=\frac{-65}{-65}=1$

$\displaystyle y=\frac{\Delta_2}{\Delta}= \frac{ \begin{vmatrix} 3 & 4 & 1 & -2\... ... -5\ 4 & 1 & 0 & -1\ 3 & 1 & -2 & -4 \end{vmatrix}}{-65}=\frac{130}{-65}=2$

$\displaystyle z=\frac{\Delta_3}{\Delta}= \frac{ \begin{vmatrix} 3 & 2 & 4 & -2\... ...& -5\ 4 & 2 & 1 & 1\ 3 & 0 & 1 & -4 \end{vmatrix}}{-65}=\frac{-195}{-65}=3$

$\displaystyle w=\frac{\Delta_4}{\Delta}= \frac{ \begin{vmatrix} 3 & 2 & 1& 4 \\... ...& 15\ 4 & 2 & 0 & 1 \ 3 & 0 & -2 & 1 \end{vmatrix}}{-65}=\frac{65}{-65}=-1$

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Cross-references: solution, column vector, column, variable, Cramer's rule, determinant, matrix

This is version 4 of example of Cramer's rule, born on 2002-07-15, modified 2005-05-23.
Object id is 3169, canonical name is ExampleOfCramersRule.
Accessed 9244 times total.

Classification:

AMS MSC:

15A15 (Linear and multilinear algebra; matrix theory :: Determinants, permanents, other special matrix functions)

Pending Errata and Addenda

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