CK-12-Pre-Calculus Concepts

http://www.ck12.org Chapter 8. Systems and Matrices

Example C

Find the determinant ofBfrom example B using Sarrus’s Rule.

Solution:

3 2 1 3 2

5 0 2 5 0

2 1 5 2 1

detB= 0 + 8 + 5 − 0 − 6 − 50 =− 43

As you can see, Sarrus’s Rule is efficient and much of the calculations can be done mentally. Additionally, zero

values make much of the multiplication easier.

Concept Problem Revisited

Determinants for 2×2 matrices are defined the way they are because of the general solution to a system of 2 variables

and 2 equations.

ax+by=e

cx+dy=f

To eliminate thex, scale the first equation bycand the second equation by a.

acx+bcy=ec

acx+ady=a f

Subtract the second equation from the first and solve fory.

ady−bcy=a f−ec

y(ad−bc) =a f−ec

y=ada f−−bcec

When you solve forxyou also getad−bcin the denominator of the general solution. This pattern led people to

start using this strategy in solving systems of equations. The determinant is defined in this way so it will always be

the denominator of the general solution of either variable.

Vocabulary

Thedeterminantof a matrix is a number calculated from the entries in a matrix. The procedure is derived from

solving linear systems.

Sarrus’s ruleis a memorization technique that enables you to compute the determinant of 3×3 matrices efficiently.

Guided Practice

1. Find the determinant of the following matrix.

C=

[− 4 12

1 − 3

]