Download A`, B`, and C`.

Chapter 14 Matrices
Objective: to find the sum, difference, or scalar multiples of matrices (14.1)
HSPA
Let u = (1, 5, -2) and v = (3, -9, 0)
1. Find u + v and u –v
2) Find 1/3v
3) Find v – 2u
4) u * v
Definitions/ Formulas
Matrix- rectangular array of numbers named by a capital letter.
Element- each number in the matrix
Dimensions- number of rows by the number of columns
Transpose of matrix- Et – interchange the rows and columns
Scalar multiplication- multiplying a matrix by a real number c
Matrix addition- adding corresponding elements (both must have the same # of rows
and columns in order to be added or subtracted)
General form of a matrix
 a11 a12
A =  a21 a22
a
 31 a32
a13 

a23 
a33 
Ex. 1 Simplify. [
−3
1
] + [ ]
0
2
3
Ex. 2 2[−4
0
0
−2
1] + [3
−1
6
3
Ex. 3 A = [0
5
4
1
−2
−1 2
3 3]
1 0
8
2) [
−1
1
3
] −[
5
4
−2
]
−1
−2
0]
11
B=
8
1
0
4
a) give the dimensions of each matrix
1
2
3
5
2
6
C=
−3
2
9
5
0
3
5
1
9
0
13
12
8
6
b) which sum is defined A + B or A + C
or B + C. Find the sum
c) give the dimensions of At
d) find At
e) At + B
f) C – At
g) Find 3C – B
Closure: Find each variable [
𝑎−3
2−𝑐
Pg 520 #7(discussion)
HW pg 521 2-16 even , 17, 19,21,
Journal#6 pg 522 # 24
−1
5
] = [
−8
4𝑏
2𝑥 + 1
]
12
Objective : to find a product of two matrices (14.2)
HSPA
1) Complete if an operation * is commutative, then a*b = _________
2) If b * h = 1/2hb , is * a commutative operation?
3) If r * h = 1/3Πr2h is * a commutative operation?
4) Find the dot product (1, 4,-3) *(-7, -2, -1).
5) Is finding a dot product of two vectors a commutative operation?
DEFINITIONS/FORMULAS
Matrix product of A* B is possible if and only if the number of columns of A equals
the number of rows of B.
You are given the dimensions of matrices A and B. Is AB defined? Is BA defined? Give
the dimensions of each possible product.
1) A3 × 5
2) A7 × 6
3) A1 × 4
B5 × 2
B2 × 7
B 3×1
4) A 2× 4
5) A3 × 4
B3 × 4
B4 × 3
Find each matrix product.
1
6) [
3
2
−5
−3
0
][0
2
1
2
4]
−1
5
7) Let A = [2 – 1 3] and B [ 0
−2
a) Is BA defined?
b) Show that (AB)t = BtAt
−1
−4]
7
8) Matrix S gives the number of three types of cars sold in March by two car dealers
and matrix P gives the profit for each type of car sold.
Dealer
1
18
[24
16
2
15
17]= S
20
profit [ $400
$650
$900] = P
Which product is defined, SP or PS? Find its matrix and interpret its elements.
Closure: The juniors at Clifton High School held a two day bake sale to raise money
for a class trip. Cupcakes, cookies, and pies were sold for $0.40, $0.25, $4.00
respectively. The first day, 84 cupcakes, 210 cookies, and 27 pies were sold. The
second day, 95 cupcakes, 184 cookies, and 17 pies were sold. Display this
information in matrix form. Then use multiplication to find the amount of money
raised each day and altogether.
HW pg 526 2-10 even, 13, 14, 16, 19, 20
Objective : to find the inverse of 2 x 2 matrix and solve linear systems using
matrices. (14.3)
HSPA
A=[
3
−4
−1
]
1
B=[
2
1
0
]
3
1. Find A * B, B * B, and A* A – B* B
Definitions/Formulas
ZERO matrix = any m × n matrix whose elements are all zeros 0m × n
Square matrix is any matrix having the same # of rows as columns.
Identity matrix (In × n ) is a matrix whose main diagonal elements are 1 and whose
other elements are 0
Additive Inverse of a matrix A denoted as – A ,is the matrix in which each element is
the opposite of its corresponding element in A.
Ex.
Multiplicative Inverse A (A-1) = I
A-1 (A) = I I= identity matrix
Inverse of 2 × 2 matrix
If Let A =
A-1 =
a b 

 . If det A ≠ 0, then A has an inverse.
c d
 d b 
1  d b 
1

=


det A  c a  ad  bc  c a 
det A = ad - cb
Find specified matrices. If the product does not exist, write not defined.
Ex. 1 M = [
3
6
4
]
7
5
N=[
−3
11
]
−7
C= [
9
12
a) N-1
b) N-1 N
c) D-1
d) - D
e) D2
f) N3
2
3
] D= [
4
5
Ex 2. Solve each matrix equation for x
a) X + 3[
2
1
12
5
]=[
0
4
7
]
−8
b) [
3
7
2
9
]X = [
5
9
3
]
1
0
7
−8
3
]
c) [
4
1
2
2
]X + [
1
6
7
2
] = [
3
0
3
]
1
Ex. 3 Write each system of equations as a single matrix equation AX = C
a) 8x – 2y = -14
b) 8x – 2y = 38
12x + 3y = 9
12x + 3y = 39
Solve by using the graphing calculator. 2x – y + z = 3
3x + 2y – 4z = 23
X – 3y – 2z = 14
HW pg. 534 1, 3, 5, 10, 12, 14, 17,18, 25
X = A-1C
Objective to solve communication network problems using matrices.(14.4)
HSPA
In the matrix below, an element “1” indicates that direct transmission from the row
computer to the column computer is possible. An element “0” indicates that direct
transmission is not possible.
To computer
A B
C
0
[1
1
1 0
1 1]
0 1
0
0
0
D
1. Which computer can send data directly to computer C?
2. Which computers can receive data directly from computer C?
3. Which computer can send data directly to the most computers?
4. Which computer can receive data directly from the most computers?
5. Which two pairs of computers have the capacity for two way direct transmission?
Definitions/ formulas
B
A
D
C
M gives the number of one-step paths
M2 gives the indirect two-step paths that use one ship as a relay.
M + M2 gives the number of paths that take either one or two steps
Ex. 1 The matrix M below models communication between computers P, Q, R, S, T.
To P Q
R
S
T
P
0 1
1
0
Q 1
1
0 1
1
0
R
0 1
0
0
1
S
1
0
0 0
1
T
1
0
0 0
0
= M
a) draw a diagram that illustrates this communication network.
b) Name the computers that can send data along the greatest number of routes via
one relay.
Classwork/HW pg 540 2, 4, 6, 9
Journal Question # 7 pg 542 #12
Objective to make predictions using powers of matrices (14.5)
HSPA
0.9
A = [0.8 0.2] T = [
0.3
0.1
]
0.7
1. Find B = AT , then C= BT and finally D= CT
2. Compare corresponding elements of matrices B, C, and D found in ex .1 Describe
what appears to be happening to the elements in each successive matrix.
Definitions/formulas
Markov Chain- a sequence of observations or predictions M0, M1, M2…. Each of which
depends on the immediately preceding one, and only on it.
A transition matrix provides a mathematical technique for analyzing, describing, and
predicting change.
Steady-state matrix- values stabilize at a certain rate.
Each year at Clifton High School, 30% of the students studying a foreign language will
not continue their studies. Also, 20% of the students are not studying a foreign
language will begin a foreign language course. Suppose 60% of the students are now
studying a foreign language.
a) Give the transition matrix T and the current population matrix P0
b) Find the percent of students studying a foreign language course next year and two
years from now.
C) show that S = [0.4 0.6] is the steady-state matrix for T.
2) a) Suppose that census takers have found that from one year to the next, 10%of
those living in a certain city will move to its suburbs while 3% of those living in the
suburbs will move into the city. Write a transition matrix T that describes the
situation.
b) If 60% of the metropolitan population is now in the city and 40% is in the suburbs,
give the current population as a 1 × 2 matrix M0.
c) Calculate M0T and interpret its elements.
d) What do the elements of M0T2 tell you?
HW pg 546 1, 2, 4,
Objective: to find images of points under different types of transformations using
matrices (14.6)
HSPA
In the figure above, point A has coordinates (r, s). Give the coordinates of the point
obtained by:
1. reflecting A in the x-axis.
2. reflecting A in the y-axis
3. translating A 2 units to the right and 3 units down.
4. rotating A 180 degrees around point O
Definitions/Formulas
Let T = [
𝑎
𝑐
𝑏
]
𝑑
represents a linear transformation that maps region R to an image region R’. Then:
1. area of R’(image)/ area R (pre-image)= absolute value of the determinant of T
2. If determinant of T is positive, then R’ and R have the same orientation.
If the determinant of T is negative, then R’ and R have opposite orientations.
If the determinant of T is zero, then R’ lies on a line through the origin.
T: (x ,y)
(ax + by, cx + dy)
T : (x , y)
(x + 2y, 3x + y)
T= [
1
3
2
]
1
x + 2y = x’
3x + y =y’
Ex. 1 Consider the transformation T: (x, y)
(2x, 3y)
a) plot points A(2,1), B(4,0), and C(1, -3) and their images A’, B’, and C’.
b) Write the transformation matrix T. Find the determinant of this matrix
c) Find the ratio of the area of
d) Compare the orientations of
A’B’C’ to the area of ABC.
ABC and
A’B’C’
Ex. 2 a) Draw
XYZ with vertices X(1, 0), Y(-1, 2) , and Z(0, 2)
b) Draw the image of XYZ under the following transformation:
T: (x, y)
(4x - 6y, 2x -3y)
c) Find the transformation matrix T and its determinant. Explain what this tells you
about the images of X, Y, Z
Ex. 3 Consider the point P(1, 2) and transformation matrices
S= [
1
4
3
]
−2
𝑇=[
4
0
1
]
5
a) Find (S◦T)(P) by first finding T(P)= P’ and then S(P’)= P”
b) Find (S◦T)(P) in a different way. ST multiply by column matrix of point P
Ex.4 Describe the motion of the translation G: (3, -1)
(8, -2) knowing that the
translation G: (x, y)
(x + h, y + k) moves every point horizontally h units and
vertically k units to its image point P’. This can be expressed by either a system of
equations or single matrix equation.
X + h = x’
Y + k = y’
Ex. 5 Express the translation G: P(x, y) ---P’(x’, y’) by matrix equation.
P(2, -4); P’(3, 1)
Book pg 556 3- 9, pg557 2, 3, 10, 13,
Journal question # 8 worksheet