section 4.1 using matrices to represent data. matrix terminology a matrix is a rectangular array of...
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Section 4.1
Using Matrices to Represent Data
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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.
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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).
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Examples of Matrices
Inventory Matrix Small
LargePicnic 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 LargePicnic tables 7⌈
9 ⌉Barbeque grills 15 12 ⌊ ⌋
⌈ 7 9 ⌉ ͟ S ͟͟ ⌈s₁₁ s₁₂ ⌉
⌊ 15 12 ⌋ ͞ ͟͞ ⌊s₂₁ s₂₂ ⌋
S is the name of the matrix.
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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.
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Solving for x and y
2x + 4 = 12Subtract 4 from both sides 2x = 8 Divide both sides by 2X = 4
-3y + 5 = 5y – 3 Add 3y to both sides5 = 8y – 3 Add 3 to both sides8 = 8yDivide both sides by 81 = y
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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.
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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 IdentityThe 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.
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Section 4.2
Matrix Multiplication
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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.
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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.
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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.
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Matrix Multiplication
⌈ 2 - 3⌉ ⌈5 0⌉Let R = | 0 5| and W = ⌊4 7⌋ 3x2 ⌊ - 2 0⌋ 2x2 Row 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)⌋
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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
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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.
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Section 4.3
The Inverse of a Matrix
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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⌉• I₃ₓ = | 0 1 0| ₃ ⌊ 0 0 1⌋
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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
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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.
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Section 4.4
Solving Systems With Matrix Equations
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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.
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Solving a Matrix Equation
Real Numbersax = b(1/a)(ax) = (1/a)(b)(1/a)(a)x = b/ax = b/a
A linear equation of the form ax = b, where a, b, and x are real numbers and a ≠ 0.
MatricesAX = BA ¹⁻ (AX) = A ¹⁻ (B)(A ¹⁻ )(A)X = A ¹(⁻ B)I X = A ¹(⁻ B)X = A ¹(⁻ B)
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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.
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Section 4.5
Using Matrix Row Operations
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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.
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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.
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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 -2R + R → R₁ ₂ ₂⌈1 1 1 : 21⌉|0 -1 -2 : -19|⌊0 1 3 : 25⌋
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Example Cont.
R + R → R₂ ₁ ₁ -1 R → R₂ ₂⌈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 R + R → R₂ ₃ ₃ R + R → R₃ ₁ ₁⌈1 0 -1 : 2 ⌉ ⌈1 0 0 : 8 ⌉|0 1 2 : 19| |0 1 2 : 19|⌊0 0 1 : 6 ⌋ ⌊0 0 1 : 6 ⌋
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Example Cont.
-2R + R → R₃ ₂ ₂⌈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
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Dependent, Independent, or Inconsistent
• Dependent System – a system of equations that has infinitely many solutions.
• Independent System – a system of equations that has exactly one solution.
• Inconsistent System – a system of equations or inequalities that has no solution.