1. 12.1 Addition of Matrices

2. Dimensions are the rows × columns Ex:
Matrix: is any rectangular array of numbers written within brackets; represented by a capital letter; classified by its dimensions
Dimensions are the rows × columns
Ex:
3 × 2
4 × 1
column matrix
2 × 2
square matrix
1 × 4
row matrix
Each number in a matrix is called an element.
We use subscripts to identify position in the matrix, aij
Ex: in matrix A, a32 is: –7

3. Properties of Matrix Addition
Two matrices are equal iff they have the same dimensions and all of their corresponding elements are equal
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 and B.
Properties of Matrix Addition
If A, B and C are m × n matrices, then
A + B is an m × n matrix
Closure
A + B = B + A
Commutative
(A + B) + C = A + (B + C)
Associative
There exists a unique m × 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
*Basically match up elements & add
Matrix Subtraction
If two matrices, A and B, have the same dimensions, then A – B = A + (–B).

4. Ex 1)
a) Find A + B
b) Find A – B
= A + (–B)
On Your Own
c) Find B – A
= B + (–A)

5. We can multiply a scalar times a matrix.
Properties of Scalar Multipication
If A, B and O are m × n matrices and c and d are scalars, then
cA is an m × n matrix
Closure
(cd)A = c(dA)
Associative
1·A = A
Multiplicative Identity
0A = O and cO = O
Multiplicative Property of the zero scalar and the zero matrix
c(A + B) = cA + cB
(c + d)A = cA + dA
Distributive Properties

6. Matrices can be used to solve many real world problems.
Ex 2) 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
Running
A = Cycling
Swimming
Running
B = Cycling
Swimming
Carl: 52 mi running, 330 mi cycling & 24 mi swimming
Flo: 66 mi running, 290 mi cycling & 16 mi swimming

7. You can also solve a “matrix equation.”
2 ways  (1) thinking algebraically & treating matrix as a whole
(2) Element by Element
(we will do both ways)
Ex 3) Solve
First distribute the 3
Method 1:
Method 2:
2x + 3 = 1
2x = –2
x = –1
2x + 6 = –2
2x = –8
x = –4
2x + 12 = 6
2x = –6
x = –3
2x + 3 = 5
2x = 2
x = 1