Download Section 8.5

8.5
APPLICATIONS OF MATRICES AND DETERMINANTS
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What You Should Learn
• Use Cramer’s Rule to solve systems of linear
equations.
• Use determinants to find the areas of triangles.
• Use a determinant to test for collinear points and
find an equation of a line passing through two
points.
• Use matrices to encode and decode messages.
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Cramer’s Rule
In this section, you will study one more method, Cramer’s
Rule, named after Gabriel Cramer.
This rule uses determinants to write the solution of a
system of linear equations.
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Cramer’s Rule
There, it was pointed out that the system
a1x + b1y = c1
a2x + b2y = c2
has a solution
and
provided that a1b2 – a2b1  0.
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Cramer’s Rule
Each numerator and denominator in this solution can
be expressed as a determinant, as follows.
Relative to the original system, the denominator for x and y
is simply the determinant of the coefficient matrix of the
system. This determinant is denoted by D.
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Cramer’s Rule
The numerators for x and y are denoted by Dx and Dy,
respectively.
They are formed by using the column of constants as
replacements for the coefficients of x and y, as follows.
Coefficient
Matrix
D
Dx
Dy
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Cramer’s Rule
For example, given the system
2x – 5y = 3
–4x + 3y = 8
the coefficient matrix, D, Dx, and Dy are as follows.
Coefficient
Matrix
D
Dx
Dy
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Cramer’s Rule
Cramer’s Rule generalizes easily to systems of n equations
in n variables. The value of each variable is given as the
quotient of two determinants.
The denominator is the determinant of the coefficient
matrix, and the numerator is the determinant of the matrix
formed by replacing the column corresponding to the
variable (being solved for) with the column representing the
constants.
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Cramer’s Rule
For instance, the solution for x3 in the following system is
shown.
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Cramer’s Rule
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Example 1 – Using Cramer’s Rule for a 2  2 System
Use Cramer’s Rule to solve the system of linear equations.
4x – 2y = 10
3x – 5y = 11
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Cramer’s Rule
Remember that Cramer’s Rule does not apply when the
determinant of the coefficient matrix is zero. This would
create division by zero, which is undefined.
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Area of a Triangle
Another application of matrices and determinants is finding
the area of a triangle whose vertices are given as points in
a coordinate plane.
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Example 3 – Finding the Area of a Triangle
Find the area of a triangle whose vertices are (1, 0), (2, 2),
and (4, 3), as shown in Figure 8.1.
Figure 8.1
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Lines in a Plane
What if the three points in Example 3 had been on the same line? What
would have happened had the area formula been applied to three such
points? The answer is that the determinant would have
been_________.
Consider, for instance, the three
collinear points (0, 1), (2, 2), and
(4, 3), as shown in Figure 8.2.
Figure 8.2
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Lines in a Plane
The result is generalized as follows.
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Example 4 – Testing for Collinear Points
Determine whether the points (–2, –2), (1, 1), and (7, 5) are
collinear. (See Figure 8.3.)
Figure 8.3
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Lines in a Plane
The test for collinear points can be adapted to another use.
That is, if you are given two points on a rectangular
coordinate system, you can find an equation of the line
passing through the two points, as follows.
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Example 5
Find an equation of the line passing through the two points
(0, 0) and (4, 5).
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Cryptography
A cryptogram is a message written according to a secret
code. (The Greek word kryptos means “hidden.”) Matrix
multiplication can be used to encode and decode
messages.
To begin, you need to assign a number to each letter in the
alphabet (with 0 assigned to a blank space), as follows.
0=_
1=A
2=B
3=C
4=D
5=E
6=F
7=G
8=H
9=I
10 = J
11 = K
12 = L
13 = M
14 = N
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Cryptography
15 = O
16 = P
17 = Q
18 = R
19 = S
20 = T
21 = U
22 = V
23 = W
24 = X
25 = Y
26 = Z
Then the message is converted to numbers and partitioned
into uncoded row matrices, each having n entries, as
demonstrated in Example 6.
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Example 6 – Forming Uncoded Row Matrices
Write the uncoded row matrices of order 1  3 for the
message MEET ME MONDAY and then encode it using
matrix
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Example 8
Use the inverse of the matrix A to decode the cryptogram:
42 88 101 88 201 251 30 33 0 26 56 64
2
2
1
A   3
7
9 
 1  4  7
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