matrix- a rectangular arrangement of numbers in rows and columns named with capital letters [ a ] ...
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Matrix- a rectangular arrangement of numbers in rows and columns
Named with capital letters [ A ] Dimensions- row x column (2 x 4)
9.2 Using Properties of Matrices
Element- each number in the matrix Named with lowercase letters and
subscripts (a₂₃) First number of subscript= row Second number of subscript=
column
Using Matrices to Represent Points/Figures in Coordinate Planes
• Use 2 row matrices to represent the vertices on a 2-dimensional figure
• 1st row = x-coordinate• 2nd row= y-coordinate• Example: Point (2,3)= 2
3
Represent figures using matricesWrite a matrix to represent the point or polygon.
a. Point A
b. Quadrilateral ABCD
Represent figures using matricesSOLUTION
a. Point matrix for A
b. Polygon matrix for ABCD
x-coordinate
y-coordinate
–4
0
x-coordinates
y-coordinates
–4 –1 4 3
0 2 1 –1
A B C D
Adding and subtracting matrices
5 15 5
0 11 –4=
6 – 1 8 –(–7) 5 – 0
4 – 4 9 – (– 2) –1 – 3=
5 + 1 –3 + 2
6 + 3 –6 + (– 4)=
6 –1
9 –10=
5 –3
6 –6
1 2
3 –4+
a.
6 8 5
4 9 –1–
1 –7 0
4 –2 3
b.
In Exercises 3 and 4, add or subtract.
3.[–3 7] + [2 –5]
SOLUTION
[–3 7] + [2 –5] =[–1 2 ]
4. 1 –4
3 –5–
2 3
7 8
SOLUTION
1 –4
3 –5–
2 3
7 8
–1 –7
–4 –13 =
Examples: Adding and subtracting matrices
Represent a translation using matrices
The matrix represents ∆ABC. Find the image 1 5 3
1 0 –1
matrix that represents the translation of ∆ABC 1 unit left and 3 units up. Then graph ∆ABC and its image.
SOLUTION
The translation matrix is
–1 –1 –1
3 3 3
Represent a translation using matrices
Add this to the polygon matrix for the preimage to find the image matrix.
–1 –1 –1
3 3 3
Translation matrix
+1 5 3
1 0 –1
Polygon matrix
A B C0 4 2
4 3 2=
Image matrix
A′ B′ C′
Multiplying matrices
Number of columns in matrix A= Number of rows in column B A X B = A B (mxn) x (nxp) (m x p)
equal1 0
4 5 Multiply
2 –3
–1 8.
SOLUTION
The matrices are both 2 X 2, so their product is defined. Use the following steps to find the elements of the product matrix.
Multiplying matrices
STEP 1
Multiply the numbers in the first row of the first matrix by the numbers in the first column of the second matrix. Put the result in the first row, first column of the product matrix.
1 0
4 5
2 –3
–1 8
1(2) + 0(–1) ?=
? ?
Multiplying matrices
1 0
4 5
2 –3
–1 8
STEP 2
Multiply the numbers in the first row of the first matrix by the numbers in the second column of the second matrix. Put the result in the first row, second column of the product matrix.
1(2) + 0(–1) 1(–3) + 0(8)=
? ?
Multiplying matrices
STEP 3
Multiply the numbers in the second row of the first matrix by the numbers in the first column of the second matrix. Put the result in the second row, first column of the product matrix.
1(2) + 0(–1) 1(–3) + 0(8)=
4(2) + 5(–1) ?
1 0
4 5 –1 8
2 –3
Multiplying matrices
STEP 4
Multiply the numbers in the second row of the first matrix by the numbers in the second column of the second matrix. Put the result in the second row, second column of the product matrix.
=4(2) + 5(–1) 4(–3) + 5(8)
1(2) + 0(–1) 1(–3) + 0(8)1 0
4 5
2 –3
–1 8
Multiplying matrices
STEP 5
Simplify the product matrix.
2 –3
3 28=
4(2) + 5(–1) 4(–3) + 5(8)
1(2) + 0(–1) 1(–3) + 0(8)
Multiplying Matrices
Use the matrices below. Is the product defined? Explain.
–3
4A =
B = [2 1] C =6.7 0
–9.3 5.2
6. AB
Yes; the number of columns in A is equal to the number of rows in B.
ANSWER
Multiplying Matrices
Use the matrices below. Is the product defined? Explain.
1. BA
ANSWER
Yes; the number of columns in B is equal to the number of rows in A.
–3
4A =
B = [2 1] C =6.7 0
–9.3 5.2
Multiplying Matrices
Use the matrices below. Is the product defined? Explain.
2. AC
ANSWER
No; the number of columns in A is not equal to the number of rows in C.
–3
4A =
B = [2 1] C =6.7 0
–9.3 5.2
Examples of Multiplying Matrices
Multiply.
1 0
0 1
3 8
–4 7
SOLUTION
=0(3) + 1(–4) 0(8) + 1(7)
1(3) + 0(–4) 1(8) + 0(7)1 0
0 1
3 8
–4 7=
3 8
–4 7
Solve a real-world problemSOFTBALL
Two softball teams submit equipment lists for the season. A bat costs $20, a ball costs $5, and a uniform costs $40. Use matrix multiplication to find the total cost of equipment for each team.
Solve a real-world problem
First, write the equipment lists and the costs per item in matrix form. You will use matrix multiplication, so you need to set up the matrices so that the number of columns of the equipment matrix matches the number of rows of the cost per item matrix.
SOLUTION
Solve a real-world problem
EQUIPMENT COST TOTAL COST
Bats Balls Uniforms Dollars Dollars
Women
MenUniforms
13 42 16
15 45 18
Bats
Balls
205
40
Women
Men
?
?=
=
Solve a real-world problem
You can find the total cost of equipment for each team by multiplying the equipment matrix by the cost per item matrix. The equipment matrix is 2 3 and the cost per item matrix is 3 1, so their product is a 2 1 matrix.
13 42 16
15 45 18
205
40
13(20) + 42(5) + 16(40)=
15(20) + 45(5) + 18(40)=
1110
1245
The total cost of equipment for the women’s team is $1110, and the total cost for the men’s team is $1245.
ANSWER