Solving Systems using Cramer's Rule

Notes, Lesson 6.5 Matrices and Determinants

We have met the operation of finding a determinant. We now find that we can find the determinant of any square matrix.

Definition:

Minor	The determinant formed when the row and column containing that element are deleted. Examples follow.
	In the matrix at the left, the minor of b is:
	In the matrix at the left, the minor of a is:
	In the matrix at the left, the minor of e is:
	In the matrix at the left, the minor of c is:
	In the matrix at the left, the minor of d is:

We speak of n-ordered determinants. Determinants can only be found for square matrices.

	Determinant of a 2nd order matrix.
	Determinant of a 3rd order matrix.
a times its minor - b times its minor + c times its minor	Another way of describing the determinant of a 3rd order matrix.

Example Problem.

Evalute this expression using minors:	Given Problem.
	Expand the expression using the definition of the determinant of a 3rd order matrix and minors.
	Find the determinants of the 2nd order matrices.
	-60 is the solution.

Thus far, we have solved linear systems by 1) Graphing; 2) Substitution; and 3) Elimination. In this lesson we will learn a 4th technique. Actually, this technique is a variation of the Elimination Method. A rule about coefficients will be developed. This rule is called Cramer's Rule. We will begin by solving a generic system of linear equations in standard form.

	Solve this given system of generic linear equations.
	Multiply equation A by e. Multiply equation B by -b. Add the equations to eliminate y. Solve for x.
	Substitute the x solution from last step into equation A. Solve for y.
	Solution
	Check the values in equation A.
	Check the values in equation B.

The values for x and y which we found above will always give us the solution to a system of two linear equations (if the equations are written in standard form). With this formula, we can ignore the x and y variables and algebraic solution techniques and concentrate on the coefficients a,b,c,d,e, and f. We can take these coefficients and put them into a matrix or grid, and concentrate on calculating with them.

Matrix	A rectangular array of numbers with columns and rows. A 2 by 2 matrix means 2 rows and 2 columns. A matrix of coefficients from the system above would be: , which is a 2 by 3 matrix.
Determinant	The determinant of a 2 by 2 matrix (Second Order Determinant) is defined by example. This operation can only be done on a square matrix.

Cramer's Rule

The solution to the system:

is:

If you look closely at Cramer's Rule above, and keep in mind the definition of a second order determinant, you will see the same results that we found at the beginning of the lesson.

Using Cramer's Rule, Solve the system:	Given Problem
a=3; b=-7; c=2; d=6; e=-13; f=4	Identify the coefficients
	Use Cramer's Rule to find x value.
	Use Cramer's Rule to find y value.
	Check Values in first equation.
	Check Values in second equation.
(2/3,0)	Solution
System is Consistent Independent	The lines intersect at one point.