12.1 Addition of Matrices

Matrix: is any rectangular array of numbers written within brackets; represented by a capital letter; classified by its dimensionsDimensions are the rows × columnsEx:

25 2

1 6 11 3 3 2 1 0

0 3 22 7

6

A B C D

3 × 2 4 × 1column matrix

2 × 2square matrix

1 × 4row 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

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 AdditionIf 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

Matrix Subtraction

If two matrices, A and B, have the same dimensions, then A – B = A + (–B).

*Basically match up elements & add

Ex 1)

a) Find A + B

5 6 7 4 2 1

1 0 2 5 1 4A B

9 8 8 5 6 7 4 2 1

4 1 2 1 0 2 5 1 4

1 4 6

6 1 6

4 2 1 5 6 7

5 1 4 1 0 2

1 4 6

6 1 6

b) Find A – B = A + (–B)

On Your Ownc) Find B – A = B + (–A)

We can multiply a scalar times a matrix.

Properties of Scalar MultipicationIf 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 = OMultiplicative Property of the zero scalar and 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.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?

WeekdayCarl Flo

Running

A = Cycling

Swimming

8 10 6 8

50 40 40 45

4 2 2 3

Running

B = Cycling

Swimming

WeekendCarl Flo

40 50 12 16 52 66

5 2 250 200 80 90 330 290

20 10 4 6 24 16

A B

Carl: 52 mi running, 330 mi cycling & 24 mi swimmingFlo: 66 mi running, 290 mi cycling & 16 mi swimming

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 1 2 1 2

2 34 1 6 5

3 6 1 2 1 42

12 3 6 5 3 1

1 2 3 62

6 5 12 3

2 82

6 2

2 8 1 41

6 2 3 12

X

X

X

X

X

Method 1:

First distribute the 3

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

Homework

#1201 Pg 600 #4, 5, 8, 10, 14, 16, 19, 20, 23, 25, 27, 29, 35, 37, 39, 43