Cramer's+Rule+for+a+3x3+System

Evan D'Elia Brian Reiff Josh Cho

Gabriel Cramer was a Swiss mathematician, born in Geneva. He was the son of physicist Jean Cramer and Anne Mallet Cramer. He showed promise in mathematics from an early age. At 18 he received his doctorate and at 20 he was co-chair of mathematics. In 1728 he proposed a solution to the St. Petersburg Paradox that came very close to the concept of expected utility theory given ten years later by Daniel Bernoulli. The famous Cramer's paradox also bears this mathematician's name but that will not be covered in this wiki.


 * Cramer's Rule** is a method in order to solve for multiple variables in an equation without using methods that would take longer like elimination or substitution. The solution is expressed in terms of the determinants of the matrix and of matrices obtained from it by replacing the variable's column with the solution column.

Given: which in matrix form is

The rules for solving a 3x3 matrix are similar to solving a 2x2 one.

First, you find the determinant of the original matrix:
 * Find the sum of the products of the down (left to right) diagonals
 * a*e*i + d*h*c + g*b*f
 * Find the sum of the products of the up (right to left) diagonals
 * g*e*c + d*b*i + a*h*f
 * Subtract the second sum from the first sum to find the determinant

Next, in order to find the values of each variable, replace the column with the coefficients of that variable with the constants (the numbers on the right-hand side of the equal sign -- shown below) and solve for those determinants.

Once you have done that, divide the determinants of the new matrices by the determinant of the original one to find the value of each variable:

The values of x, y, and z are:



@http://www.college-cram.com/study/algebra/systems-of-equations/cramers-rule-3x3/


 * So as an example** if we have a three variable system of equations equation such as:

2x + 3y + z = 10 x - y + z = 4 4x - y - 5z = -8

First we arrange the matrix:

2 3 1 1 -1 1 4 -1 -5

Second, we find the determinant for this matrix:

[2(-1)(-5) + 1(-1)(1) + 4(3)(1)] - [4(-1)(1) + 1(3)(-5) + -1(1)(2)] = [10 + (-1) + 12] - [(-4) + (-15) + (-2)] = 21 - (-21) =
 * 42**

Now we can begin to get the determinants for each variable by replacing each variable's column with the answer column

10 3 1 X = 4 -1 1 -8 -1 -5

2 10 1 Y = 1 4 1 4 -8 -5

2 3 10 Z = 1 -1 4 4 -1 -8 ...

X = 84 Y = 42 Z = 126

To find the value of each variable we divide its determinant by the determinant of the original matrix

x = 84/42 y = 42/42 z = 126/42


 * x = 2**
 * y = 1**
 * z = 3**

We can check these answers by plugging them back into our system of equations (below is an example of just one equation but you can plug these number into any of the original equations and it will work out):

2(2) + 3(1) + (3) =? 10

4 + 3 + 3 =? 10


 * 10 = 10 (correct)**

Program

This is a program where you can solve systems of equations using Cramer's Law. Simply type in the coefficients for the variables and the answers and solve any system!

Click Open when the pop-up window comes up


 * For more practice problems, check out Period 6's Cramer's Rule 3x3 Matrix page.**