They are related in a sense that they describe the charachteristic of an equation.Quote:

Originally Posted byQuick

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There are 3 types systems involving linear equations: inconsistent, consistent and dependent and consistent and independent.

Inconsistent: If you are given two equations that are contradictory there are no solutions. For example,

$\displaystyle \left\{ \begin{array}{c}x+y=1\\x+y=2$

How can two numbers add to 1 and 2? So there are no solutions. But I want you to look at is determinant of its coefficient.

$\displaystyle \left| \begin{array}{cc}1&1\\1&1 \end{array} \right| = 0$. It is zero.

Consistent and Dependant. This is a system that has infinitely many solutions. For example,

$\displaystyle \left\{ \begin{array}{c}x+y=1\\2x+2y=2 \end{array} $

Because the second equation can be divided by two to give,

$\displaystyle x+y=1$ which is the same as the first. Thus for any value of $\displaystyle x$ you can find a $\displaystyle y$. Thus, there are infinitely many solutions.

Now, find the determinat of the coefficients.

$\displaystyle \left| \begin{array}{cc}1&1\\2&2 \end{array} \right|=0$. Also, zero.

WarningWhen you find the determinant of the system you need the variables to belined up. For example,

$\displaystyle \left\{ \begin{array}{c}2x+y+z=1\\2y+x+z=2\\2z+x+y=3$. You need to line up the x's the y's and the z'sand thentake the determinant. Thus, it is,

$\displaystyle \left| \begin{array}{ccc}2&1&1\\1&2&1\\1&1&2 \end{array} \right| $

Consistent and Independant. This is a system with auniquesolution. For example,

$\displaystyle \left\{ \begin{array}{c}x+2y=1\\2x+y=1 \end{array}$.

Its determinant of the coefficients is,

$\displaystyle \left| \begin{array}{cc}1&2\\2&1 \end{array} \right|=3\not = 0 $. This leads to the following theorem.

TheoremA system of linear equations is consistent and independant (unique solution) only when, its determinant is N0N-ZER0.

Of course, when the determinant is zero it can have no solutions (inconsistent) or infinitely many solutions (consistent and dependant). However, how can you determine which one it is when determinant is zero? Later on I can show you.

DefinitionA system of linear equations,

$\displaystyle \left\{ \begin{array}{c}{ a_{11}x_1+a_{12}x_2+...+a_{1n}x_n=0\\a_{21}x_1+a_{ 22}x_2+...+a_{2n}x_n=0\\.......................... ........=0\\a_{n1}x_1+a_{n2}x_2+...+a_{nn}x_n=0 $ is calledhomogenous.

TheoremIf a homogenous system of linear equations has its determinant zero then it has infinitely many solutions.

ProofWhen the determinant of the system

$\displaystyle \left| \begin{array}{cccc}a_{11}&a_{12}&...&a_{1n}\\a_{21 }&a_{22}&...&a_{2n}\\...&...&...&...\\a_{n1}&a_{n2 }&...&a_{nn}\end{array} \right| = 0$

Then it can have either infintely many solution or no solutions (it cannot have a unique solution). But this equation does have solution, namely thetrivial,

$\displaystyle x_1=x_2=...=x_n=0$

Thus, it CANNOT have no solutions, thus it must have infinitely many solutions. (This type of equation does appear in applied math often and it is the case when the determinat is zero because otherwise it is not interesting).

Hope you like this lecture, that should entertain you for some hours. Next, time I will show you how to solve equations with determinants..