Chapter SevenLinear Systems and Matrices

7.7 Determinants7.8 Applications of Determinants

Ch. 7 Overview

• Solving Systems of Equations• Systems of Linear Equations in Two

Variables• Multivariable Linear Systems• Matrices and Systems of Equations

Ch. 7 Overview (cont.)

• Operations with Matrices• The Inverse of a Square Matrix• The Determinant of a Square Matrix• Applications of Matrices and

Determinants

7.7 – The Determinant of a Square Matrix

• The Determinant of a 2X2 Matrix

• The Determinant of a Square Matrix

• Triangular Matrices

DETERMINANTS

Determinants are mathematical objects (scalars) that are very useful in the analysis and solution of systems of linear equations. As shown byCramer's rule, a system of linear equations has a unique solution iff thea determinant of the system's matrix is nonzero For example, eliminating x, y, and z from the equations

gives the square coefficient matrix with a unique number which is calledthe determinant for this system of equation. Determinants are defined only for square matrices. If the determinant of a matrix is 0, the matrix is said to be singular, (has no solution)

1 2 3

1 2 3

1 2 3

000

a x a y a zb x b y b zc x c y c z

7.7 – The Determinant of a 2X2 Matrix

The determinant of the matrix is a scalar.

is given by det A = |A| = ad – cb.

a bDet A

c d

Example

Find the determinant of the matrix:

2 93 6

A

15

Example-You Try

Find the determinant of the matrix:

5

2 13 4

A

Example

Find the following determinant using the diagonal method or the minors and cofactors method:

0 2 13 1 24 0 1

14

Example- You Try

Find the following determinant by hand:

34

1 2 04 0 61 3 5

7.7 – Triangular Matrices

If you have either an upper-triangular, lower-triangular or a diagonal matrix there is a really easy way to find the determinant:

Multiply the entries on the main diagonal.

Example

Find the following determinant:

2 0 0 04 2 0 05 6 1 0

1 5 3 3

12

7.8 – Applications of Matrices and Determinants

• Area of a Triangle

• Test for Collinear Points

Area of a Triangle

For a triangle with vertices:

Area = 1 1

2 2

3 3

11 12

1

x yx yx y

Make sure that area is always positive!

1 1 2 2 3 3, , , , ,x y x y x y

Example

• Find the area of the triangle with vertices:

• Draw it

1,0 , 2,2 , 4,3

3 square units2

Example

Find the area of the triangle with vertices:

• Draw it

3,5 , 2,6 , 3, 5

28 square units

Test for collinear points

are collinear if:

1 1

2 2

3 3

11 01

x yx yx y

1 1 2 2 3 3, , , , ,x y x y x y

Example

Determine whether the point are collinear:

0,1 , 4,4 , 8,7yes, they are

Can you think of another way to tell?

Example-You Try

Determine whether the point are collinear:

no, they are not

2, 2 , 1,1 , 7,5

Example

Find x such that the triangle has an area of 4 square units.

4,2 , 3,5 , 1, x

x=3, x=19

Example

Use a determinant to find the equation of a line through (3,5) & (-2,3).

2 195 5

y x

1 1

2 2

11 01

x yx yx y

You Try

Use a determinant to find the equation of a line through (2,1) & (-5,8).

3y x

1 1

2 2

11 01

x yx yx y

Homework 7.7 & 7.8

7.7 page 533 3-11 odd, 31,33, 51

7.8 page 544 1-13 odd, 25