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3/30 Ch. 12A Quiz
4/4 Ch. 12A TEST – part graphing calc + no calc
BRING graphing calculators starting TOMORROW!
Lesson 12 – 1 Addition
of Matrices
PRE-CALCULUS
Pg. 600 #4, 5, 8, 10, 14, 16, 19, 20, 23, 25, 27, 29,
35, 37, 39, 43
Learning Objective
 To add matrices
Matrix
Matrix: is any rectangular array of numbers
written within brackets; represented by a
capital letter; classified by its dimensions
Dimensions are the rows x columns
2
5
2
𝐵= 1
𝐴 = −1 3
0
2 −7
6
3𝑥2
4𝑥1
Column matrix
𝐶=
6
−3
−1
2
2𝑥2
Square matrix
𝐷= 3
2 1
0
1𝑥4
Row matrix
Each number in a matrix is called an element.
Matrix
We use subscripts to identify position in the
matrix, 𝑎𝑖𝑗
5
2
1. In matrix 𝐴 = −1 3 , what is 𝑎32 ?
2 −7
𝑎32  3rd row
 2nd column
𝑎32 = −7
Matrix
Two matrices are equal iff they have the same
dimensions and all of their corresponding
elements are equal
Matrix Addition
Don’t WRITE!!
YOU HAVE THIS!!
If two matrices, A and B have the same
dimensions, then their sum 𝐴 + 𝐵 is a matrix
of the same dimensions whose elements are
the sums of the corresponding elements of A &
B.
*Basically match up elements & add
Matrix Subtraction
Don’t WRITE!!
YOU HAVE THIS!!
If two matrices, A and B have the same
dimensions, then 𝐴 − 𝐵 = 𝐴 + (−𝐵)
Properties of Matrix Addition
Matrix
Don’t WRITE!!
YOU HAVE THIS!!
If 𝐴, 𝐵, 𝑎𝑛𝑑 𝐶 are 𝑚 𝑥 𝑛 matrices, then
𝐴 + 𝐵 is an 𝑚 𝑥 𝑛 matrix
Closure
𝐴+𝐵 =𝐵+𝐴
Commutative
𝐴 + 𝐵 + 𝐶 = 𝐴 + (𝐵 + 𝐶) Associative
There exists a unique 𝑚 𝑥 𝑛
matrix 𝑂 such that
𝑂+𝐴 =𝐴+𝑂 =𝐴
Additive
Identity
For each 𝐴, there exists a
unique matrix −𝐴 such that
𝐴 + −𝐴 = 𝑂
Additive
Inverse
5 6
Given: 𝐴 =
1 0
Matrix
7
2
4 2
𝐵=
−5 1
1
−4
2. Find 𝐴 + 𝐵
5+4
6+2
1 + (−5) 0 + 1
𝐴+𝐵 =
9
−4
8 8
1 −2
7+1
2 + (−4)
5 6
Given: 𝐴 =
1 0
Matrix
3. Find 𝐴 − 𝐵
5
=
1
7
2
4 2
𝐵=
−5 1
= 𝐴 + (−𝐵)
−4 −2 −1
6 7
+
5 −1 4
0 2
5 + (−4) 6 + (−2)
7 + (−1)
1+5
2+4
=
1
6
0 + (−1)
4 6
−1 6
1
−4
5 6
Given: 𝐴 =
1 0
Matrix
4. Find 𝐵 − 𝐴
=
4
−5
7
2
= 𝐵 + (−𝐴)
2 1
−5
+
1 −4
−1
4 + (−5)
−1
−6
−6 −7
0 −2
2 + (−6) 1 + (−7)
−5 + (−1) 1 + 0
=
4 2
𝐵=
−5 1
−4 −6
1 −6
−4 + (−2)
1
−4
Properties of Scalar Multiplication
Matrix
If 𝐴, 𝐵, 𝑎𝑛𝑑 𝑂 are 𝑚 𝑥 𝑛 matrices and 𝑐 and 𝑑
are scalars, then
𝑐𝐴 is an 𝑚 𝑥 𝑛 matrix
Don’t WRITE!!
YOU HAVE THIS!!
𝑐𝑑 𝐴 = 𝑐(𝑑𝐴)
Closure
Associative
1∙𝐴 =𝐴
Multiplicative
Identity
𝑂𝐴 = 𝑂 and 𝑐𝑂 = 𝑂
Multiplicative
Property of the zero
scalar & the zero
matrix
𝑐 𝐴 + 𝐵 = 𝑐𝐴 + 𝑐𝐵
𝑐 + 𝑑 𝐴 = 𝑐𝐴 + 𝑑𝐴
Distributive
Properties
Matrices can be used to solve many real world problems.
Matrix
Change to #5!!!!
On half sheet.
Not #2.
5. Carl & Flo are training for a triathlon by running,
cycling, & swimming. The matrices below show the
number of miles that each devotes to each activity, both
on weekdays & weekend days. What is the total number
of miles that each devotes to each activity in a 7 – day
week?
Weekend
Weekday
Carl
6
Run
𝐵 = Cycle 40
Swim 2
Carl
8
Run
𝐴 = Cycle 50
Swim 4
Flo
10
40
2
40
5𝐴 + 2𝐵 = 250
20
12 16
50
200 + 80 90
4
6
10
Total
Carl Flo
Run
52
66
Cycle = 330 290
Swim
24
16
Flo
8
45
3
Carl: 52 mi run, 330 mi
cycle, 24 mi swim
Flo: 66 mi run, 290 mi
cycle, 16 mi swim
You can also solve a “matrix equation”
2 ways  (1) Thinking algebraically & treating
matrix as a whole
Matrix
(2) Element by Element
6. Solve 2𝑋 + 3
2
+
3
12
6
1
=
3
6
−2
5
1
4
1
2
=
6
1
−2
5
First distribute the 3
3 6
1 −2
2𝑋 +
=
12 3
6 5
Method 1:
1 −2
3 6
2𝑋 =
−
6 5
12 3
−2 −8
2𝑋 =
−6 2
1 −2 −8
−1 −4
𝑋=
𝑋=
2 −6
2
−3 1
You can also solve a “matrix equation”
2 ways 
Matrix
(2) Element by Element
6. Solve 2𝑋 + 3
2𝑥
2𝑥
2𝑥
3
+
2𝑥
12
6
1
=
3
6
−2
5
1
4
1
2
=
6
1
−2
5
First distribute the 3
3 6
1 −2
2𝑋 +
=
12 3
6 5
Method 2:
2𝑥 + 3 = 1
𝑥 = −1
2𝑥 + 12 = 6
𝑥 = −3
2𝑥 + 6 = −2
𝑥 = −4
2𝑥 + 3 = 5
𝑥=1
𝑋=
−1 −4
−3 1
Assignment
Pg. 600 #4, 5, 8, 10, 14, 16, 19, 20, 23, 25, 27, 29,
35, 37, 39, 43