review test c
TRANSCRIPT
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A Quick Method for Finding the
Multiplicative Inverse of a 2 x 2
Matrix
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Example
Find the multiplicative inverse of A using the Quick
Method to find the inverse.
3 2A=
1 4
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Example
Find the multiplicative inverse of A using the
Quick Method. Check your work using your
2 3calculator. A=
1 5
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Example
Show that A does not have an inverse. First by calculations,
then use your calculator and see what you get for an answer.
3 2A=
6 4
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Finding Multiplicative Inverses of n x
n Matrices with n Greater Than 2
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If ad-bc=0 then the matrix has no multiplicative
inverse.
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Example
1 1
2 2
Find the inverse function without a calculator for2 1
A= . Show that A A I and A A=I .1 3
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Find the multiplicative inverse of A by row calculations,
then check your work using the calculator.
1 1 0
A= 1 3 4 .
0 4 3
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Example
Find the multiplicative inverse matrix of A using row
calculations. Then check your answer using your calculator.
1 2 2
A= 0 1 1 .2 1 0
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Solving Systems ofEquations Using
Multiplicative Inverses
of Matrices
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1Solve the system using A , the inverse of the coefficient matrix.
x+z=3
x-y=-2
x-y+2z=2
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Example1
Solve the system by using A , the inverse of thecoefficient matrix.
x+ y- z =2
2y+ z=3
2 1x y
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Applications of Matrix Inverses to
Coding
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A cryptogram is a message written so that no one other than the intended reci-
pient can understand it. To encode a message, we begin by assigning a number
to each letter in the alphabet: A=1, B=2, C=3, . . .Z=26, and a space =0. The
numerical equivalent of the word ATTITUDE=1,20,20,9,20,21,4,5
The numerical equivalent of the word MATH is 13,1,20,8. The numerical
equivalent of the message is then converted into a matrix. Finally, an
invertible matrix can be used to convert the message into code. The
multiplicative inverse of this matrix can be used to decode the message.
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Encoding the Word MATH
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Decoding a Word
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Example
1
4 1For the word CASH which is 3,1,19,8, use the coding matrix A=
3 1
1 1to encode the word. Then use the matrix A to decode the given
3 4
word. The problem has already been started for you
!
.
4 1 3 19
3 1 1 8
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(a)
(b)
(c)
(d)
1 3Find the multiplicative inverse of A= .
2 2
2 3
2 1
1 0
0 11 3
4 8
1 1
4 81 3
2 4
1 1
2 4
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(a)
(b)
(c)
(d)
1 3 1Find the multiplicative inverse of A= .
2 2 1
2 3 1
2 1 1
1 0 20 1 1
1 4 1
1 2 0
No inverse exists
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Example
Evaluate the determinant of each of the following matices:
2 3.
5 1
3 2.
4 1
a
b
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Example
Use Cramer's Rule to solve the system:
2x-3y=-11
x+2y=12
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Example
Use Cramer's Rule to solve the system:
3x+2y=-1
2x-4y=10
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The Determinant of a
3 x 3 Matrix
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Evaluate the determinant of the following matrix:
2 1 0
1 1 2
3 1 0
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Example
Evaluate the determinant by hand, then check youranswer on the calculator.
2 1 3
3 0 1
1 2 3
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Solving Systems of Linear Equations
in Three Variables Using
Determinants
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Example
Use Cramer's rule to solve:
-2x+y =1
x-y-2z=2
3x+y =6
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Cramers Rule with Inconsistent and
Dependent Systems
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The Determinant of Any
N x N Matrix
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The determinant of a matrix with n rows and n columns is said
to be an nth-order determinant. The value of an nth-order determinant
can be found in terms of determinants of order n-1.
We can generalize
the idea for fourth-order determinants and
higher. We have seen that the minor of the element a is the
determinant obtained by deleting the ith row and the jth columnin the given array of numbers
i j
. The cofactor of the element a
is (-1) times the minor of the a entry. If the sum of the row
and column (i+j) is even, the cofactor is the same as the minor.
If the sum of the row and column
i j
i j
ijth
(i+j) is odd, the cofactor is theopposite of the minor.
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Example
Evaluate the determinant of the following matrix. Noticethat you can use either the third or the fourth columns.
1 2 0 0
0 1 2 0
1 2 0 11 3 1 1
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(a)
(b)
(c)
(d)
Evaluate the determinant
3 2
1 4 -
14
10
8
11
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(a)
(b)
(c)
(d)
Use Cramer's Rule to solve the linear systems.
-x+2y=7
2x-2y=-4
( 1, 2)
(2, 2)
(3,4)
(3,5)