Review Test C

8/3/2019 Review Test C

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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)

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Review Test C