1. Lesson 12 – 1 Addition of Matrices
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

2. Learning Objective
To add matrices

3. Matrix: is any rectangular array of numbers written within brackets; represented by a capital letter; classified by its dimensions
Matrix
Dimensions are the rows x columns 𝐵=
𝐴= 5 2 −1 3 2 −7
3 𝑥 2
4 𝑥 1
Column matrix
𝐶= 6 −1 −3 2 𝐷=
2 𝑥 2
1 𝑥 4
Square matrix
Row matrix

4. Matrix Each number in a matrix is called an element.
We use subscripts to identify position in the matrix, 𝑎 𝑖𝑗
1. In matrix 𝐴= 5 2 −1 3 2 −7 , what is 𝑎 32 ?
𝑎 32  3rd row
𝑎 32 =−7
 2nd column

5. Matrix Don’t WRITE!! YOU HAVE THIS!! Don’t WRITE!! YOU HAVE THIS!!
Two matrices are equal iff they have the same dimensions and all of their corresponding elements are equal
Matrix
Matrix Addition
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.
Don’t WRITE!! YOU HAVE THIS!!
*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)

6. Matrix Don’t WRITE!! YOU HAVE THIS!! Properties of Matrix Addition
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
Don’t WRITE!! YOU HAVE THIS!!
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

7. Matrix Given: 𝐴= 5 6 7 1 0 2 𝐵= 4 2 1 −5 1 −4 2. Find 𝐴+𝐵 5+4 6+2 7+1
1+(−5)
0+1
2+(−4)
𝐴+𝐵= −4 1 −2

8. Matrix Given: 𝐴= 5 6 7 1 0 2 𝐵= 4 2 1 −5 1 −4 3. Find 𝐴−𝐵 =𝐴+(−𝐵)
= −4 −2 −1 5 −1 4
5+(−4)
6+(−2)
7+(−1)
1+5
0+(−1)
2+4
= −1 6

9. Matrix Given: 𝐴= 5 6 7 1 0 2 𝐵= 4 2 1 −5 1 −4 4. Find 𝐵−𝐴 =𝐵+(−𝐴)
= −5 1 −4 + −5 −6 −7 −1 0 −2
4+(−5)
2+(−6)
1+(−7)
−5+(−1)
1+0
−4+(−2)
= −1 −4 −6 −6 1 −6

10. Matrix Don’t WRITE!! YOU HAVE THIS!!
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
Closure
Don’t WRITE!! YOU HAVE THIS!!
𝑐𝑑 𝐴=𝑐(𝑑𝐴)
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

11. Matrix Change to #5!!!! On half sheet. Not #2. Weekday Carl Flo
Matrices can be used to solve many real world problems.
Matrix
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?
Weekday
Carl Flo
Weekend
Carl Flo
Change to #5!!!!
On half sheet. Not #2. 𝐴=
Run
Cycle
Swim 𝐵=
Run
Cycle
Swim =
5𝐴+2𝐵
Total
Carl Flo
Carl: 52 mi run, 330 mi cycle, 24 mi swim
Run
Cycle
Swim =
Flo: 66 mi run, 290 mi cycle, 16 mi swim

12. Matrix You can also solve a “matrix equation” 2 ways 
(1) Thinking algebraically & treating matrix as a whole
(2) Element by Element
6. Solve 2𝑋 = 1 −2 6 5
First distribute the 3
2𝑋 = 1 −2 6 5
= 1 −2 6 5
Method 1:
2𝑋= 1 − −
2𝑋= −2 −8 −6 2
𝑋= −2 −8 −6 2
𝑋= −1 −4 −3 1

13. Matrix You can also solve a “matrix equation” 2 ways 
(2) Element by Element
6. Solve 2𝑋 = 1 −2 6 5
First distribute the 3
2𝑋 = 1 −2 6 5
2𝑥 2𝑥 2𝑥 2𝑥 = 1 −2 6 5
Method 2:
2𝑥+3=1
2𝑥+6=−2
𝑥=−1
𝑥=−4
2𝑥+12=6
2𝑥+3=5
𝑋= −1 −4 −3 1
𝑥=−3
𝑥=1

14. Assignment
Pg. 600 #4, 5, 8, 10, 14, 16, 19, 20, 23, 25, 27, 29, 35, 37, 39, 43