12.1 addition of matrices. matrix: is any rectangular array of numbers written within brackets;...
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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