Transcript Algebra 2 Section 4-4 Matrices and Determinants What You’ll Learn Why It’s Important    To evaluate the determinant of a 3 x 3 matrix, To find the.

Algebra 2
Section 4-4
Matrices and Determinants
What You’ll Learn
Why It’s Important



To evaluate the determinant of a 3 x 3 matrix,
To find the area of a triangle, given the
coordinates of its vertices
You can use matrices and determinants to
solve problems involving geometry and
geography
Evaluating a third-order determinant using
Diagonals

In this method, you begin by writing the first two
columns on the right side of the determinant.
aa b
c
dd e
gg h
f
i
Evaluating a third-order determinant using
Diagonals

Next, draw diagonals from each element of the top
row of the determinant downward to the right. Find
the product of the elements on each diagonal.
a
b
c
a
b
d
g
e
h
f
i
d
g
e
h
aei + bfg + cdh
Evaluating a third-order determinant using
Diagonals

Then, draw diagonals from the elements in the third
row of the determinant upward to the right. Find the
product of the elements on each diagonal.
a
b
c
a
b
d
g
e
h
f
i
d
g
e
h
gec + hfa + idb
Evaluating a third-order determinant using
Diagonals

To find the value of the determinant, add the
products of the first set of diagonals and then
subtract the sum of the products of the second set
of diagonals.
(aei+ bfg + cdh)
-
(gec+ hfa + idb)
Example 1
1 0 8

Evaluate
7
2
3 4
2 5
using diagonals.
Solution Example 1

First, rewrite the first two columns to the right
of the determinant.
-11 0 8
7
2
2
3
4
22 5
Solution Example 1

Next, find the products of the elements of the
diagonals (going down).
1 0 8 1 0
-1(3)(5) + 0(4)(2)+ 8(7)(2)
7
3 4
7
3
-15 + 0 + 112
2
2 5
2
2
97
Solution Example 1

Next, find the products of the elements of the
diagonals (going up).
1 0 8 1 0
7
3 4
7
3
2
2 5
2
2
2(3)(8) + 2(4)(-1) + 5(7)(0)
48 - 8 + 0
40
Solution Example 1

To find the value of the determinant, add the
products of the first set of diagonals (going
down) and then subtract the sum of the
products of the second set of diagonals
(going up).
-1(3)(5) + 0(4)(2) + 8(7)(2)
-15 + 0 + 112
97
97 - 40
57
2(3)(8) + 2(4)(-1) + 5(7)(0)
48 - 8 + 0
40
What is the purpose of determinants?


One very powerful application of determinants is
finding the areas of polygons.
The formula below shows how determinants serve
as a mathematical tool to find the area of a triangle
when the coordinates of the three vertices are given.
Area of
Triangles
The area of a triangle having vertices
at (a,b), (c,d), and (e,f) is |A|, where
Notice that it is necessary to use
the absolute value of A to
guarantee a nonnegative value
for area
a
A
1
c
2
e
b 1
d
1
f
1
Example 2

Find the area of the triangle whose vertices
are located at (3,-4), (5,4), and (-3,2)
Solution Example 2
a
A
1
c
2
e

Find the area of the triangle whose vertices are
located at (3,-4), (5,4), and (-3,2)

Assign values to a, b, c, d, e, and f and substitute
them into the area formula and evaluate.



(a,b) = (3,-4)
(c,d) = (5,4)
(e,f) = (-3,2)
3
1
A 5
2
3
4 1
4
2
1
1
b 1
d
1
f
1
Evaluate the
determinant
and then
multiply by ½
Solution Example 2
3 4 1
1
A 5
4 1
2
3 2 1
3
1
A 5
2
3
4 1 3 -4
4
2
3(4)(1) + (-4)(1)(-3) + (1)(5)(2)
1 5 4
1 -3 2
12 + 12 + 10
34
-3(4)(1) + 2(1)(3) + (1)(5)(-4)
A = ½(60)
A = 30 Square Units
-12 + 6 - 20
-26
34 - (-26) = 60