4.3 Matrices and Determinants

Algebra 2

Learning Targets:

• Evaluate the determinant of a 3 x 3 matrix, and

• Find the area of a triangle given the coordinates of its vertices.

Determinants• Every square matrix has a

determinant. The determinant has the same elements as the matrix, but they are enclosed between vertical bars instead of brackets. In Chapter 3, you learned a method for evaluating a 2 x 2 determinant.

Determinants• The determinant of

is . To evaluate

The determinant, use the rule for second order determinants.

1117

231117

23

67)34(33

)17)(2()11(31117

23

or

bcaddc

ba

Expansion by Minors• A method called expansion by minors can be used to

evaluate the determinant of a 3 x 3 matrix. The minor of an element is the determinant formed when the row and column containing that element are deleted. For the determinant

65

28

659

284

731

1min

.24

71

659

284

731

5min,

659

284

731

orisorofthe

orisorofthe

Expansion by Minors

• To use expansion by minors with third-order determinants, each member of one row is multiplied by its minor. The signs of the products alternate, beginning with the second product. The definition below shows an expansion using the elements in the first for of the determinant. However, ANY row can be used.

hg

edc

ig

fdb

ih

fea

ihg

fed

cba

Expansion of a Third-Order Determinant

Ex. 1: Evaluate the determinant of Using expansion by minors.

821

756

432

43

2812352

)512(4)748(3)1440(2

21

564

81

763

82

752

821

756

432

hg

edc

ig

fdb

ih

fea

ihg

fed

cba

Using Diagonals• Another method for evaluating a third

order determinant is using diagonals. • STEP 1: You begin by repeating the first

two columns on the right side of the determinant.

h

e

b

g

d

a

ihg

fed

cba

ihg

fed

cba

Using Diagonals• STEP 2: Draw a diagonal from each element in

the top row diagonally downward. Find the product of the numbers on each diagonal.

h

e

b

g

d

a

ihg

fed

cba

ihg

fed

cba

aei bfg cdh

Using Diagonals• STEP 3: Then draw a diagonal from each

element in the bottom row diagonally upward. Find the product of the numbers on each .

h

e

b

g

d

a

ihg

fed

cba

ihg

fed

cba

idbhfagec

Using Diagonals• To find the value of the determinant, add

the products in the first set of diagonals, and then subtract the products from the second set of diagonals.

The value is: aei + bfg + cdh – gec – hfa – idb

Ex. 2: Evaluate using diagonals.

1

2

4

3

3

1

213

523

041

213

523

041

First, rewrite the first two columns along side the determinant.

Ex. 2: Evaluate using diagonals.

1

2

4

3

3

1

213

523

041

213

523

041

Next, find the values using the diagonals.

4 60 0

0 -5 24

Now add the bottom products and subtract the top products.

4 + 60 + 0 – 0 – (-5) – 24 = 45. The value of the determinant is 45.

Area of a triangle• Determinants can be used to find the

area of a triangle when you know the coordinates of the three vertices. The area of a triangle whose vertices have coordinates (a, b), (c, d), (e, f) can be found by using the formula:

,

1

1

1

2

1

fe

dc

ba

A and then finding |A|, since the area cannot be negative.

Ex. 3: Find the area of the triangle whose vertices have coordinates (-4, -1), (3, 2), (4, 6).

How to start: Assign values to a, b, c, d, e, and f and substitute them into the area formula and evaluate.

,

1

1

1

2

1

fe

dc

ba

A a = -4, b = -1, c = 3, d = 2, e = 4, f = 6

6

2

1

4

3

4

164

123

114

2

1

A

-8 -4 18

8 -24-3

Now add the bottom products and subtract the top products.

-8 + (-4) + 18 – 8 – (-24) –(-3) = 25. The value of the determinant is 25. Applied to the area formula ½ (25) = 12.5. The area of the triangle is 12.5 square units.

Ex. 4: Solve for n if582

334

722

634

n

n

Sometimes one or more of the elements of a determinant may be unknown, but the value of the determinant is known. You can use expansion to find the values of the variable.

24n – 84 – 36n + 48n +84n – 18 = -582 120n – 102 = -582 120n = -480 n = -4The value of n is -4.

n

n

n

n

3

2

3

4

2

4

334

722

634

Collect like terms

Add 102 to both sides

Divide by 120 both sides.

Assignment

• pp. 218 #12-34 Even n9os