section 4 4 matrices.pdfsolving for x and y 2x + 4 = 12 subtract 4 from both sides 2x = 8 divide...
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Matrix Terminology
• A matrix is a rectangular array of numbers enclosed in a single set of brackets.
• The dimensions of a matrix are the number of horizontal rows and the number of vertical columns it has.
• For example, if a matrix has 2 rows and 3 columns, its dimensions are 2 x 3.
• Each number in the matrix is an called an entry or element.
Using Matrices to Represent Data
Inventory (June 1)Small Large
Sales (June)Small Large
Deliveries (June)Small Large
Picnic tables
8 10 7 9 15 20
Barbeque grills
15 12 15 12 18 24
The table below shows business activity for one month in a home-improvement store. The table shows stock (inventory on June 1), sales (during June), and receipt of new goods (deliveries in June).
Examples of Matrices
Inventory Matrix
Small Large
Picnic tables ⌈ 8 10 ⌉
Barbeque grills ⌊ 15 12 ⌋
⌈ 8 10 ⌉ ͟ M ͟ ⌈m₁₁ m₁₂⌉
⌊ 15 12 ⌋ ͞ ͞ ⌊m₂₁ m₂₂⌋
M is the name of the matrix.
Sales Matrix
Small Large
Picnic tables ⌈ 7 9 ⌉
Barbeque grills ⌊ 15 12 ⌋
⌈ 7 9 ⌉ ͟ S ͟ ⌈s₁₁ s₁₂⌉
⌊ 15 12 ⌋ ͞ ͞ ⌊s₂₁ s₂₂⌋
S is the name of the matrix.
If Two Matrices are Equal
• Two matrices are equal if they have the same dimensions and if corresponding entries are equivalent.
• Solve ⌈2x+4 5 1⌉ ͟ ⌈12 5 1⌉ for x & y.
⌊ -2 -3y + 5 -4⌋ ͞ ⌊-2 5y – 3 -4⌋
Because the matrices are equal:
2x + 4 = 12 and -3y + 5 = 5y – 3.
Solving for x and y
2x + 4 = 12
Subtract 4 from both sides
2x = 8
Divide both sides by 2
X = 4
-3y + 5 = 5y – 3
Add 3y to both sides
5 = 8y – 3
Add 3 to both sides
8 = 8y
Divide both sides by 8
1 = y
Addition and Scalar Multiplication
• To find the sum (or difference) of matrices A and B with the same dimensions, find the sums (or differences) of corresponding entries in A and B.
• Scalar multiplication is multiplication of each entry in a matrix by the same real number.
Properties of Matrix Addition
• For matrices A, B, and C, each with dimensions of m x n:
• Commutative A + B = B + A• Associative (A + B) + C = A + (B + C)• Additive Identity The m x n matrix having 0 as
all of its entries is the m x n identity matrix for addition.
• Additive Inverse For every m x n matrix A, the matrix whose entries are the opposite of those in A is the additive inverse of A.
Matrix Multiplication
• If matrix A has dimensions m x n and matrix B has dimensions n x r, then the product AB has dimensions m x r.
• Find the entry in row i and column j of AB by finding the sum of the products of the corresponding entries in row i of A and column j of B.
Matrix Multiplication
• If matrix A has the dimensions 2 x 3 and matrix B has the dimensions 3 x 2, then:
• AB will have the dimensions 2 x 2.
• BA will have the dimensions 3 x 3.
• For AB, 2 x 3 3 x 2, the 2’s are the outer dimensions and the 3’s are the inner dimensions.
• For BA, 3 x 2 2 x 3, the 3’s are the outer dimensions and the 2’s are the inner dimensions.
Matrix Multiplication
• If the inner dimensions are the same, then multiplication can occur. The outer dimensions give the product dimensions after multiplication occurs.
• If the inner dimensions are not the same, then multiplication cannot occur.
Matrix Multiplication
⌈ 2 - 3⌉ ⌈5 0⌉
Let R = | 0 5| and W = ⌊4 7⌋
3x2 ⌊ - 2 0⌋ 2x2Row 1 of R, Column 1 of W Row 1 of R, Column 2 of W
⌈(2)(5) + (-3)(4) (2)(0) + (-3)(7)⌉
Row 2 of R, Column 1 of W Row 2 of R, Column 2 of W
|(0)(5) + (5)(4) (0)(0) + (5)(7)|
Row 3 of R, Column 1 of W Row 3 of R, Column 2 of W
⌊(-2)(5) + (0)(4) (-2)(0) + (0)(7)⌋
Matrix Multiplication
⌈ - 2 - 21⌉
RW = | 20 35|
⌊ - 10 0⌋
WR - does not exist because the inner dimensions do no not match.
W 2 x 2 3 x 2 R
Matrix Multiplication
• A network is a finite set of connect points.
• Each point is called a vertex.
• A directed network is a network in which permissible directions of travel between the vertices are indicated.
• You can represent a network in an adjacency matrix, which indicates how many one-stage (direct) paths are possible from one vertex to another.
Square Matrix
• A square matrix is a matrix that has the same number of columns and rows. 2x2, 3x3, 4x4,…
• An identity matrix, called I, has 1’s on the main diagonal and 0’s elsewhere.
⌈ 1 0 0⌉
• I₃x₃ = | 0 1 0|
⌊ 0 0 1⌋
More About Matrices
The Identity Matrix of Mult.
• Let A be a square matrix with n rows and n columns. Let I be a matrix with the same dimensions and with 1’s on the main diagonal and 0’s elsewhere. Then AI = IA = A
The Inverse of a Matrix
• Let A be a square matrix with n rows and n columns. If there is an n x n matrix B, such that AB = I, then A and B are inverses of one another. The inverse of matrix A is denoted by A⁻ ¹.
• Note A⁻ ¹ ≠ 1/A
Determinant of a 2 x 2 Matrix
• Let A = ⌈a b⌉.
⌊c d⌋
The determinant of A, denoted by det(A) or
|a b|
|c d|,
Is defined as det(A) = |a b| = ad – bc.
|c d|
Matrix A has an inverse if and only if det(A) ≠ 0.
Solving Systems With Matrix Equations
• A matrix equation – an equation of the form AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
• A system of linear equations can be used to represent situations and be written as a matrix equation.
Solving a Matrix Equation
Real Numbers
ax = b
(1/a)(ax) = (1/a)(b)
(1/a)(a)x = b/a
x = b/a
A linear equation of the form ax = b, where a, b, and x are real numbers and a ≠ 0.
Matrices
AX = B
A⁻ ¹(AX) = A⁻ ¹(B)
(A⁻ ¹)(A)X = A⁻ ¹(B)
I X = A⁻ ¹(B)
X = A⁻ ¹(B)
Example
5x + 2y – z = - 7 ⌈5 2 - 1⌉ ⌈x⌉ ⌈- 7⌉
x - 2y + 2z = 0 → |1 - 2 2||y| = | 0 |
3y + z = 17 ⌊ 0 3 1⌋ ⌊z⌋ ⌊17⌋
⌈x⌉ ⌈5 2 - 1⌉⁻ ¹ ⌈- 7⌉
|y| = |1 - 2 2| | 0 |
⌊z⌋ ⌊ 0 3 1⌋ ⌊17⌋
⌈x⌉ ⌈- 2⌉
|y| = | 4 |
⌊z⌋ ⌊ 5 ⌋ Thus, the solution is x = - 2, y = 4, and z = 5.
Using Matrix Row Operations
• The row-reduction method of solving a system allows you to determine whether the system is independent, dependent, or inconsistent.
• The row-reduction method is performed on an augmented matrix. An augmented matrix consists of the coefficients and constant terms in the system of equations.
• Reduced row-echelon form – an augmented matrix is in this form if the coefficient columns form an identity matrix.
Elementary Row Operations
• The following operations produce equivalent matrices, and may be used in any order and as many times as necessary to obtain reduced row-echelon form.
• Interchange two rows.
• Multiply all entries in one row by a nonzero #.
• Add a multiple of one row to another.
Example
System Augmented Matrixm + a + n = 21 ⌈1 1 1 : 21⌉
2m + a = 23 |2 1 0 : 23|
a + 3n = 25 ⌊0 1 3 : 25⌋
coefficients constants
-2R₁ + R₂ → R₂
⌈1 1 1 : 21⌉
|0 -1 -2 : -19|
⌊0 1 3 : 25⌋
Example Cont.
R₂ + R₁ → R₁ -1 R₂ → R₂
⌈1 0 - 1: 2 ⌉ ⌈1 0 - 1: 2 ⌉
|0 -1 -2 : -19| |0 1 2 : 19|
⌊0 1 3 : 25⌋ ⌊0 1 3 : 25⌋
-1 R₂ + R₃ → R₃ R₃ + R₁ → R₁
⌈1 0 -1 : 2 ⌉ ⌈1 0 0 : 8 ⌉
|0 1 2 : 19| |0 1 2 : 19|
⌊0 0 1 : 6 ⌋ ⌊0 0 1 : 6 ⌋
Example Cont.
-2R₃ + R₂ → R₂
⌈1 0 0 : 8 ⌉
|0 1 0 : 7 |
⌊0 0 1 : 6 ⌋
The matrix is now in reduced row-echelon form.
m = 8, a = 7, n = 6