matrices - lancaster high school · 2 matrix addition and subtraction definition: if two matrices a...
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Matrices The dimensions of a matrix are written number of rows by number of columns.
In 1 – 3 give the dimensions of the matrix.
1)
823
472 2)
529
534
972
851
3)
10
12
Each number in the brackets is called an element of the matrix. In example 2 above, the
4 is the element in row 3 column 1.
4) According to the Census Bureau, in 1900 the median age at first marriage was 25.9 for males and 21.9 for females. In 1930, the median age at first marriage was 24.3
for males and 21.3 for females. In 1960, it was 22.8 for males and 20.3 for females. In 1990, it was 26.1 for males and 23.9 for females. Store this information in a 2 x 4
matrix.
5) The matrix at the right gives data from the
1992 Statistical Abstract of the United States
regarding prisoners executed under civil
authority from 1930 to 1989.
a) Give the dimensions of this matrix.
b) What does the entry in row 2, column 1 represent?
c) How many people were executed in the period 1970 – 1979?
2
Matrix Addition and Subtraction Definition: If two matrices A and B have the same dimensions, their sum A + B is the
matrix in which each element is the sum of the corresponding elements in A and B.
Example 1: Example 2:
141812
182015
643
875
+
534
465
321
423
=
987
654
321
+
987
654
321
=
Definition: If two matrices A and B have the same dimensions, their difference A - B is the
matrix in which each element of B is subtracted from the corresponding element in A.
Example 1: Example 2:
141812
182015
643
875
-
534
465
321
423
=
534
465
321
423
-
141812
182015
643
875
=
Scalar Multiplication
Matrix addition is related to a special multiplication involving matrices called scalar
multiplication. Consider this repeated addition:
24
87 +
24
87 +
24
87 =
Notice that in the final result, every element of the original matrix has been multiplied by 3. With real numbers, we use multiplication as a shorthand for repeated addition; for
example, the sum 7 + 7 + 7 can be written as 3 x 7. Similarly, we can rewrite the above
sum as:
24
873
The constant 3 is called a scalar. Scalar Multiplication is the product of a scalar k and a
matrix A is the matrix kA in which each element is k times the corresponding element in A.
Find the product of:
−
1194
1275 =
3
Using Matrices to Solve Systems
Write the following systems of equations as a matrix:
1)
=−
=−
20y10x8
9y4x3 2)
−=+
−=+−
−=−+
8yx
36zyx5
11z7y4x3
Write a system of equations for each and then write the system as a matrix:
3) At a local video rental store John rents two movies and three games for a total of
$15.50. At the same time, Meg rents three movies and one game for a total of $12.05. How much money is needed to rent a combination of one game and one
movie?
4) The total attendance at a school play was 1, 250. The cost of tickets was $6 for
students and $7.50 for adults. The school drama club had a revenue total of
$8, 362.50. How many of each ticket was sold for the play?
5) Joe bought two new compact disks and 3 used compact disks for $54. At the same
prices, Susan bought three new compact disks and one used compact disk for $53. Find the cost of buying a new and a used compact disk.
6) Yolanda, Raphael, and Tim go to Bonzo Burger. Yolanda gets 1 burger, 1 fry, 2
sodas (she’s thirsty) and her total is $6.90. Raphael gets 2 burgers, 2 fries, 1 soda and spends $10. Tim gets 3 burgers, 1 fry, and 1 soda, which totals $10.70. What is the
price of one soda?
4
Using x, y and z (when necessary) as your variables write a system of equations given the
following matrices:
1)
−
1973
2814
8423
2)
−
−
1413
228
3)
−
410
221
4) Using number 3 above what is the value of y?
Find x.
5)
−
3100
6210
9531
6) Using number 5 above what is the value of z?
Find x and y.
5
Using Row-Echelon Form to Solve Matrices A matrix in Row-Echelon form has
� All rows consisting entirely of zero in the bottom matrix.
� For each row that does not consist entirely of zeros, the first non-zero entry is
1.(leading 1)
� For two successive (nonzero) rows the leading 1 in the higher row is further to the left
than the leading 1 in the lower row.
Determine if the following matrices are in row-echelon form:
1)
−
1973
2814
8423
2)
−
410
221
3)
−
0000
6210
9531
4)
−
−
1413
228
5)
−
3100
6210
9532
In order to get a matrix in row-echelon form you need to apply row operations. 1) Interchange any two rows. ex. R1 ↔ R2
2) Replace a row by a nonzero multiple of that row. ex. 2R1
3) Replace a row by the sum of that row and a constant nonzero multiple of
another row. ex. –2R1 + R3
Use row operations to put the following matrix in row-echelon form:
3125
2242
6
1) Solve the system:
=+−
=+−
=+−
17z5y5x2
4y3x
9z3y2x
2) Solve the system:
=+
=−
12y6x3
16y4x2
3) Solve the system:
=++
=++
−=−+
1zy2x3
0z2y2x2
4z2y3x5
7
4) Solve the system:
=−
=+−
=++
3zx3
8zyx2
9z3y2x
5) Solve the system:
−=−+
=+−
=−+
1z3yx2
0zy2x
1zy2x3
8
6) John, Paul, and Mike go to Burger Land. John gets 5 burgers, 2 fries and 3 apple pies
for a total of $14.50. Paul gets 3 burgers, 1 fry and 2 apple pie for $8.50. Mike gets 1
Burger, 1 fry and 1 apple pie which totals $4.50 (what a light-weight!). What is the price
of one burger?
7) Solve the system:
−=−−−
−=−++
=−+
−=−+
19wz7y4x
2w3zy4x2
2zy2x
3w2zy
10
Number of Solutions
The number of Solutions of a Linear System
For a system of linear equations, exactly one of the following is true.
1) There is exactly one solution.
2) There are infinitely many solutions.
3) There is no solution.
A system of linear equations is called consistent if it has at least one solution. A consistent
system with exactly one solution is independent. A consistent system with infinitely many
solutions is dependent. A system of linear equations is called inconsistent if it has no
solutions.
Directions: For the following matrices determine whether they are consistent or
inconsistent. If it is consistent determine if it is independent or dependent.
1)
−
4000
2810
8421
2)
−
410
221
3)
−
0000
6210
9531
4)
1411
000
5)
−
3100
6210
9531
6)
323
700
13
Matrix Multiplication Row By Column Multiplication
In linear-combination applications, it is useful to store data in matrices.
Consider a movie theater that charges $6 for adults over 17, $4 for students 13 – 17, and
$2.50 for children 12 or under. How much does it cost a family with 2 adults, 1 student, and
3 children to enter the theater?
The answer is $6•2 + $4•1 + $2.5•3 = $23.50. This is the same arithmetic needed to
calculate the product of these matrices:
[ ]
•
3
1
2
5.246
cost per category • number in each category
Multiplying Two Matrices
In general, the product A••••B or AB of two matrices A and B is found by multiplying the rows
of matrix A by the columns of matrix B.
**Matrices can only be multiplied when the number of columns for the left matrix equals
the number of rows for the right matrix.
These matrices can be multiplied: These matrices cannot be
multiplied:
−
14
28
240
531 =
22164
36168
240
531
−
14
28
2 x 2 2 x 3 2 x 3 2 x 3 2 x 2
equal not equal
dimensions of product
In general, multiplication of matrices is not commutative.
Example: [ ]
•
3
1
2
5.246 =
Example: Let A =
−
14
28 and B =
240
531. Find the product AB.
14
1) [ ]1102 −
−
−
0
1
0
7
0
1
1
5
2
0
1
1
2) [ ]
−−
0
2
0
1
0
2
1
3
4
213
3)
−
1021
1102
−
−
0
1
0
7
0
1
1
5
2
0
1
1
4)
−
− 31
12
10
11
5) Suppose you sell 3 T-shirts at $10 each, 4 hats at $15 each, and 1 pair of shorts at $20.
Set up 2 matrices in order to find the total revenue.
15
Identity Matrix
1) What number is the multiplicative identity? Why?
2) What is the additive identity? Why?
3) Fill in the matrix to make the following statement true:
24
87
??
?? =
24
87
4) Fill in the matrix to make the following statement true:
987
654
321
???
???
???
=
987
654
321
16
Definition of Identity Matrix
The n x n matrix that consists of 1’s on its main diagonal and 0’s elsewhere is called the
identity matrix of order n and is denoted by
=
10000
0...100
0...010
0...001
In
MMMMM
If A is an n x n matrix then A nI =A.
Inverse of a Square Matrix
1) What is the multiplicative inverse of 5? Why?
2) What is the additive inverse of 5? Why?
Definition of the Inverse of a Square Matrix Let A be an n x n matrix and let In be the n x n identity matrix. If there exists a matrix A-1
such that
AA-1 = In = A-1A
then A-1 is called the inverse of A.
1) Show that B is the inverse of A, where A =
−
−
11
21 and B =
−
−
11
21
17
Determinant
For a 2 x 2 matrix the determinant can be calculated by
det [M] = (upper left times lower right) – (upper right times lower left)
det
dc
ba = (ad) – (bc)
det
78
23 = (3)(7) – (2)(8) = 5
Find the determinant of the for each of the following:
1)
−
31
12 2)
− 63
58
If matrix M =
dc
ba then
−
−
ac
bd is called the adjoint of [M]. Find the adjoint matrix for
each of the following.
3)
− 52
61 4)
86
42
18
Finding the Inverse
Inverse: The inverse of [ ]M is [M] 1− = ]M[adj]Mdet[
1⋅
Find the inverse of the following matrices.
1)
−− 31
42 2)
−− 11
12
3)
− 52
61 4)
86
43
19
Solving Using the Inverse
We can use inverses to solve a system of linear equations.
If A is an invertible matrix, the system of linear equations represented by AX = B has a
unique solution
X = A 1− B
Ex) Use an inverse matrix to solve the following systems.
1)
=+
=−
12y6x3
16y4x2
2)
=−
=−
2y3x2
4y2x
20
Cramer’s Rule
To solve a system of linear equations using Cramer’s Rule, x = det
det x , y = det
dety,…
Just watch. You’ll get it.
Use Cramer’s Rule to solve the following systems.
1)
=−
=−
11y5x3
10y2x4
2)
=+
−=+
4y3x5
2y4x3
3)
=+−
=++
−=+−
1z6y2x5
10z3y2x2
5zyx4
21
Cryptography
A cryptogram is a message written according to a secret code. 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).
To encode a message, choose an n x n invertible matrix A and multiply the uncoded row
matrices by A (on the right) to obtain coded row matrices.
Use the following matrix to encode the message MEET ME MONDAY.
A =
−−
−
−
411
311
221
To decode the message, we simply multiply the coded row matrices by A 1− (on the right).