“no one can be told what the matrix is… they have to see it for themselves…” - lawrence ...
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“No one can be told what the matrix is… they have to see it for themselves…” - Lawrence Fishburne. It’s a good thing we have Section 7.2a!!!. Definition: Matrix. Let m and n be positive integers. An m x n matrix (read “ m by n matrix”) is a rectangular array of m rows - PowerPoint PPT PresentationTRANSCRIPT
““No one can be told what the matrix is…No one can be told what the matrix is…they have to see it for themselves…”they have to see it for themselves…”
- Lawrence Fishburne- Lawrence Fishburne
It’s a good thing we have Section 7.2a!!!It’s a good thing we have Section 7.2a!!!
Definition: Matrix
Let m and n be positive integers. An m x n matrix(read “m by n matrix”) is a rectangular array of m rowsand n columns of real numbers.
We also use the shorthand notation [a ] for this matrix.ij
11 12 1
21 22 2
1 2
n
n
m m mn
a a a
a a a
a a a
Definition: Matrix
Each element, or entry, a , of the matrix uses double subscriptnotation. The row subscript is the first subscript i, and thecolumn subscript is j. The element a is in the i-th row andj-th column. In general the order of an m x n matrix is m x n.If m = n, the matrix is a square matrix. Two matrices are equalmatrices if they have the same order and their correspondingelements are equal.
ij
ij
Practice Problem!!!
1 2 3
2 0 4
Determine the order of the matrix, and identify the specifiedelements.
12a
21a
Order 2 x 3Order 2 x 3
= –2= –2
= 2= 2
Practice Problem!!!
1 1
0 4
2 1
3 2
Determine the order of the matrix, and identify the specifiedelements.
22a
43a
Order 4 x 2Order 4 x 2
= 4= 4
Doesn’t exist!Doesn’t exist!
Matrix Addition and Subtraction
We can add or subtract matrices of the same order by addingor subtracting their corresponding entries. Matrices of differentorders cannot be added or subtracted!!!
Let and be matrices of order m x n.A ija B ijb
A B ij ija b
A B ij ija b
1. The sum A + B is the m x n matrix
2. The difference A – B is the m x n matrix
Scalar Multiplication of MatricesWhen dealing with matrices, real numbers are scalars. Theproduct of the real number k and the m x n matrixis the m x n matrix
A ija
The matrix is a scalar multiple of A.
A ijk ka
A ijk ka
A Few More Definitions…
Let A = [a ] be any m x n matrix. The m x n matrix O = [0]consisting entirely of zeros is the zero matrix becauseA + O = A. In other words, O is the additive identity for theset of all m x n matrices. The m x n matrix B = [–a ] consistingof the additive inverses of the entries of A is the additiveinverse of A because A + B = O.
ij
ij
Practice Problem!!!
1 0 2
A 4 1 1
2 0 1
For the given matrices, find (a) A + B, (b) A – B, (c) 3A, and(d) 2A – 3B.
2 1 0
B 1 0 2
4 3 1
1 1 2
A B 3 1 1
6 3 0
3 1 2
A B 5 1 3
2 3 2
Practice Problem!!!
1 0 2
A 4 1 1
2 0 1
For the given matrices, find (a) A + B, (b) A – B, (c) 3A, and(d) 2A – 3B.
2 1 0
B 1 0 2
4 3 1
3 0 6
3A 12 3 3
6 0 3
8 3 4
2A 3B 11 2 8
8 9 5
Practice Problem!!!Let A = [a ] and B = [b ] be 2 x 2 matrices with a = 3i – j andb = i + j – 3 for i = 1, 2, and j = 1, 2.
2 1A
5 4
ij ijij
ij2 2
1. Determine A and B. 1 2B
2 5
Practice Problem!!!Let A = [a ] and B = [b ] be 2 x 2 matrices with a = 3i – j andb = i + j – 3 for i = 1, 2, and j = 1, 2.
2 1A
5 4
ij ijij
ij2 2
2. Determine the additive inverse –A of A and verify that A + (–A) = [0]. What is the order of [0]?
2 1A
5 4
0 0A A 0
0 0
The order of [0] is 2 x 2.
Practice Problem!!!Let A = [a ] and B = [b ] be 2 x 2 matrices with a = 3i – j andb = i + j – 3 for i = 1, 2, and j = 1, 2.
ij ijij
ij2 2
3. Determine 3A – 2B. 8 13A 2B
11 2
MatrixMultiplication
Definition: Matrix Multiplication
Let A = [a ] be an m x r matrix and B = [b ] an r x n matrix.ij ij
The product AB = [c ] is the m x n matrix whereij
1 1 2 2ij i j i j ir rjc a b a b a b
To multiply two matrices, the columns of the first matrix must equal the rows of the second matrix. The resulting matrix has rows and columns determined by the “outside” values.
Ex: Can we multiply a 3 x 2 matrix and a 2 x 4 matrix???
(3 x 2)(2 x 4)(3 x 2)(2 x 4)Yes, we can multiply…Yes, we can multiply…and the result is a 3 x 4 matrix…and the result is a 3 x 4 matrix…
Find the product AB, where possible:
2 1 3A
0 1 2
1 4
B 0 2
1 0
1 6AB
2 2
Support with a calculator???Support with a calculator???
2 1 1 0 3 1 2 4 1 2 3 0
0 1 1 0 2 1 0 4 1 2 2 0AB
Find the product AB, where possible:
2 1 3A
0 1 2
3 4B
2 1
The product AB is not defined!!! Why??The product AB is not defined!!! Why??
A florist makes three different cut flower arrangements (I, II, andIII). Matrix A shows the number of each type of flower used ineach arrangement.
5 8 7
6 6 7
4 3 3
1.50 1.35
0.95 1.00
1.30 1.35
The florist can buy his flowers from two different wholesalers(W1 and W2), but wants to give all his business to one or theother. The cost of the three flower types from the two whole-salers is shown in matrix B.
I II IIIRoses
A = Carnations
Lilies
Roses
B = Carnations
Lilies
W1 W2
5 8 7
6 6 7
4 3 3
1.50 1.35
0.95 1.00
1.30 1.35
Construct a matrix showing the cost of making each of the threeflower arrangements from flowers supplied by the two differentwholesalers.
I II IIIRoses
A = Carnations
Lilies
Roses
B = Carnations
Lilies
W1 W2
We want the columns of A to match up with the rows of B, so wefirst switch the rows and columns of A:
5 6 4
8 6 3
7 7 3
Rose Carn LilyI
II
III
The new matrix is called thetranspose of A, and is denoted AT
5 8 7
6 6 7
4 3 3
1.50 1.35
0.95 1.00
1.30 1.35
Construct a matrix showing the cost of making each of the threeflower arrangements from flowers supplied by the two differentwholesalers.
I II IIIRoses
A = Carnations
Lilies
Roses
B = Carnations
Lilies
W1 W2
Now, we find the product A B:
5 6 4
8 6 3
7 7 3
Rose Carn LilyI
II
III
T
1.50 1.35
0.95 1.00
1.30 1.35
Rose
Carn
Lily
W1 W2
x
18.40 18.15
21.60 20.85
21.05 20.50
I
= II
III
W1 W2