9-5 Higher Order Determinants

Cramer’s Rule and 3 by 3’s

How to Evaluate a 3 by 3 Determinant

a b cd e fg h i

a b c a bd e f d eg h i g h

aei bfg cdh

+ +

ceg

afh bdi

– ––

Example1. Evaluate

3 1 21 0 3 4 2 1

Cramer’s RuleIt will look scary at first, but it is the SAME

EXACT CONCEPT as a 2 by 2 situation.

That is, the denominator of each solution is the determinant of the coefficients.

The numerator is the coefficient determinant with the appropriate column replaced by the solutions. That is, the x column replaced when solving for x, the y column when solving for y and the z column when solving for z.

1 1

2 2

3 3

1 1 1

2 2 2

3 3 3

1

2

3

b cb cb c

x a b ca b ca b c

ddd

1 1 1

2 2 2

3 3

1

3

2

3

a x b y c za x b y c za x b y c z

ddd

1

2

1 1

2 2

3 3

1 1 1

2 2

3 3 3

3

2

a ca ca c

y a b ca b ca b c

ddd

1 1

2 2

3 3

1 1 1

2 2

2

3

3 3

1

2

3

a ba ba b

z a b ca b ca b c

ddd

Hint: if you can find x and y, just sub in to find z

Examples2.

3.

x 2y z 32x y z 4x y 2z 5

5x 2y 2z 52x 3y 5z 13x 2y 3z 4

Uhoh…what happens when the denominator equals zero?