Cramer

Iris Aliaj · Poincare Conjecture Offline Joined: 02 Jun 2004 Posts: 165

What is Cramer's Rule? How is it used in solving systems? What do the determinants in Cramer's method for solving systems represent?

P.S.: I don't know where to post this topic. Maybe the moderators will move it to another section.

**Posted:** Fri Aug 20, 2004 3:37 am

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liyi

i think this belongs to Advanced Algebra section.

You can find it in almost every linear algebra book.
You can also have a look at
http://mathworld.wolfram.com/CramersRule.html

**Posted:** Fri Aug 20, 2004 4:59 am

amcavoy · Yang-Mills Theory Offline Joined: 21 Mar 2005 Posts: 922

I know this is in the wrong section, but I didn't want to begin a new thread. Maybe this can be moved.

Cramer's Rule states the following:

$x_i=\frac{\det A_i(\mathbf{x})}{\det A}$

Here's an example:

$x_1+2x_2=5$
$3x_1+7x_2=17$

I will compute $x_1$ .

$A=\begin{bmatrix}1 & 2 \\ 3 & 7\end{bmatrix}$

The meaning of $A_i$ is that you replace the "i"th column of A with the vector x, which in this case is <5,17>.

$A_1=\begin{bmatrix}5 & 2 \\ 17 & 7\end{bmatrix}$

So $\det A=7-6=1$ and $\det A_1(\mathbf{x})=35-34=1$ .

$x_1=\frac{\det A_1(\mathbf{x})}{\det A}=1$

Maybe you can try to find $x_2$ with this method. It is actually very simple and quick. I can provide a proof of the rule as well if you need it.

Alex.

**Posted:** Thu Jul 28, 2005 5:29 pm

boxedexe

Cramer's Rule is: If we have a set of equations
$\\a_1x+b_1y+c_1z=k_1\\a_2x+b_2y+c_2z=k_2\\a_3x+b_3y+c_3z=k_3$
$x=\frac{\left|\begin{matrix}k_1&b_1&c_1\\k_2&b_2&c_2\\k_3&b_3&c_3\end{matrix}\right|}{\Delta}$ $y=\frac{\left|\begin{matrix}a_1&k_1&c_1\\a_2&k_2&c_2\\a_3&k_3&c_3\end{matrix}\right|}{\Delta}$ $z=\frac{\left|\begin{matrix}a_1&b_1&k_1\\a_2&b_2&k_2\\a_3&b_3&k_3\end{matrix}\right|}{\Delta}$ $\Delta=\left|\begin{matrix}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{matrix}\right|$

Of course, this also works for higher order determinants.

P.S. I'm 14 years old, so I don't know if this is around what you wanted to know. Smile

Masoud Zargar

**Posted:** Sat Oct 15, 2005 10:06 pm

Frozen

What if our system has coeficients not in $R$ (or $C$ ), but in a field $K$ ? If $K$ is commutative, it is clear, Cramer is still available. But what if it is not commutative?

**Posted:** Tue Feb 28, 2006 1:40 pm

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ZetaX

Zero'th problem: fields are commutative for me Wink

First problem for noncommutative case:
A linear equation is not necessary of type $a_1 x_1 + a_2 x_2 +...=d$ , but can contain more than one linear term of type $a_1 x_1 b_1$ with $x_1$ .

Next problem: the definition of determinant looses it's symmetry of changing rows or columns, thus even interchanging the equations or variables could give us another determinant.

Third problem: even the division in the rule is not defined on which side to be taken.

Conclusion: by some random constants, e.g. from the quaternions, there should be a counterexample.

PS: moderators, please move this thread...

**Posted:** Tue Feb 28, 2006 3:19 pm

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