Solve Systems with Matrices - Amazon S3
Transcript of Solve Systems with Matrices - Amazon S3
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Solve Systems with MatricesCollege Algebra
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Describing Matrices
A matrix is a rectangular array of numbers. A row in a matrix is a set of numbers that are aligned horizontally. A column in a matrix is a set of numbers that are aligned vertically. Matrices are enclosed in or , and are usually named with capital letters.Examples:
π΄ = 1 23 4 , π΅ =
1 2 70 β5 67 8 2
, πΆ =β1 30 23 1
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Describing Matrices
A matrix is often referred to by its size or dimensions: πΓπ indicating πrows and π columns. Matrix entries are defined first by row and then by column. For example, the entry π34 is located at row π, column π.
π΄ =π77 π78 π79π87 π88 π89π97 π98 π99
A square matrix has dimensions πΓπ, meaning it has the same number of rows and columns.A row matrix has dimensions 1Γπ (one row), such as π΅ = π77 π78 π79
A column matrix has dimensions πΓ1 (one column), such as πΆ =π77π87
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Adding and Subtracting Matrices
Given matrices π΄ and π΅ of like dimensions, addition and subtraction of π΄and π΅ will produce matrix πΆ or matrix π· of the same dimension.
π΄ + π΅ = πΆ such that π34 + π34 = π34π΄ β π΅ = π· such that π34 β π34 = π34
Example:
π΄ = β2 30 1 and π΅ = 8 1
5 4π΄ + π΅ = 6 4
5 5π΄ β π΅ = β10 2
β5 β3
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Multiplying a Matrix by a Scalar
Scalar multiplication involves finding the product of a constant by each entry in the matrix.
Given π΄ =π77 π78π87 π88 , the scalar multiple ππ΄ =
ππ77 ππ78ππ87 ππ88
Scalar multiplication is distributive. For the matrices π΄ and π΅ with scalars πand π,
π π΄ + π΅ = ππ΄ + ππ΅π + π π΄ = ππ΄ + ππ΄
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Product of Two Matrices
The product of two matrices is only possible when the inner dimensions are the same. If π΄ is an πΓπ matrix and π΅ is an πΓπ matrix, the product matrix π΄π΅ is an πΓπ matrix.To obtain the entry in row π, column π of π΄π΅, multiply row π in π΄ by column πin π΅ as follows:
π37 π38 π3@ Aπ74π84πB4
= π37π74 + π38π84 + β―+ π3@πB4
Matrix multiplication is associative: π΄π΅ πΆ = π΄(π΅πΆ)Matrix multiplication is distributive: πΆ π΄ + π΅ = πΆπ΄ + πΆπ΅
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Product of Two Matrices
Example: Multiply π΄ = 1 2 34 5 6 and π΅ =
7 108 119 12
Solution: Since π΄ has dimension 2Γ3 and π΅ has dimension 3Γ2, π΄π΅ has dimension 2Γ2.
π΄π΅ = 1 7 + 2 8 + 3(9) 1 10 + 2 11 + 3(12)4 7 + 5 8 + 6(9) 4(10) + 5 11 + 6(12)
π΄π΅ = 50 68122 167
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Systems of Equations and Matrices
A matrix can be used to represent and solve a system of equations. The coefficients of the variables and the constants become the entries in a matrix. A vertical line separates the coefficient entries from the constants; this is called an augmented matrix.Example:
G3π₯ β π¦ β π§ = 0π₯ + π¦ = 52π₯ β 3π§ = 2
is represented as3 β1 β11 1 02 0 β3
|052
Notice that all the variables line up in their own columns, and missing terms have a coefficient of 0.
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Row Operations and Row-Echelon Form
In order to solve the system of equations, we convert the augmented matrix to row-echelon form in which there are ones down the main diagonal, and zeros in every position below the main diagonal.
1 π π0 1 π0 0 1
We use row operations to obtain a new matrix that is row-equivalent.1. Interchange rows. (Notation: π 3 β π 4)2. Multiply a row by a constant. (Notation: ππ 3)3. Add the product of a row multiplied by a constant to another row.
(Notation: π 3 + ππ 4)
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Gaussian Elimination
The Gaussian elimination method refers to a strategy to obtain the reduced row-echelon form of a matrix.
Example: 1 23 β2|
10β2
β3π 7 + π 8 = π 81 20 β8|
10β32 Obtain a zero in row 2, column 1
β7Oπ 8 = π 8
1 20 1|
104 Obtain a one in row 2, column 2
β2π 8 + π 7 = π 71 00 1|
24 Obtain a zero in row 1, column 2
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Gaussian Elimination
Given a system of equations, create the augmented matrix from the coefficients and constants. Reduce the matrix to row-echelon form to obtain the solution for the system.For example:
P π₯ + 2π¦ = 103π₯ β 2π¦ = β2 is written as 1 2
3 β2|10β2
Reduce this matrix to 1 00 1|
24
Rewrite the system as Pπ₯ = 2π¦ = 4 for the solution of 2,4
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Identity Matrix and Multiplicative Inverse
The identity matrix, πΌ@, is a square matrix containing ones down the main diagonal and zeros everywhere else.
πΌ8 =1 00 1 πΌ9 =
1 0 00 1 00 0 1
If π΄ is an πΓπ matrix and π΅ is an πΓπ matrix such that π΄π΅ = π΅π΄ = πΌ@, then π΅ = π΄S7, the multiplicative inverse of matrix π΄.
Example: π΄ = 1 5β2 β9 , π΅ = β9 β5
2 1
π΄π΅ = 1 β9 + 5(2) 1 β5 + 5(1)β2 β9 β 9(2) β2 β5 β 9(1) = 1 0
0 1 = πΌ8
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Find the Inverse Using Matrix Multiplication
To find the inverse of a given matrix, multiply it by a matrix containing unknown constants and set it equal to the identity. This will result in systems of equations that can be used to find the unknowns.
Example: π΄ = 1 β22 β3
1 β22 β3 A π π
π π = 1 00 1 Set up π΄ A π΄S7 = πΌ
1π β 2π 1π β 2π2π β 3π 2π β 3π = 1 0
0 1 Find the product
P1π β 2π = 12π β 3π = 0 , P1π β 2π = 0
2π β 3π = 1 Create systems and solve for π, π and π, π
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Find the Inverse by Augmenting with the Identity
When matrix π΄ is transformed into πΌ, the augmented matrix πΌ transforms into π΄S7.
Example: π΄ = 2 15 3
2 15 3|
1 00 1 Augment π΄ with the identity
1 00 1|
3 β1β5 2 Perform row operations to turn π΄ into the identity
π΄S7 = 3 β1β5 2 Inverse is the right side of the augmented matrix
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Inverse of a 2Γ2 Matrix
If π΄ is a 2Γ2 matrix such as π΄ = π ππ π , the multiplicative inverse of π΄ is
given by the formulaπ΄S7 =
1ππ β ππ
π βπβπ π
If ππ β ππ = 0, then π΄ has no inverse.
Example: π΄ = 1 β22 β3
Solution: π΄S7 = 77 S9 S(S8)(8)
β3 2β2 1 = β3 2
β2 1
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Solve a System of Equations Using an Inverse
Given a system of equations, write the coefficient matrix π΄, the variable matrix π, and the constant matrix π΅. Then π΄π = π΅. Multiply both sides by the inverse of π΄ to obtain the solution, π = π΄S7π΅.
Example: Solve the system of equations P 3π₯ + 8π¦ = 54π₯ + 11π¦ = 7
3 84 11
π₯π¦ = 5
7 Write the system in matrix terms
π΄S7 = 11 β8β4 3 Use the formula for the inverse of a 2Γ2 matrix
π₯π¦ = 11 β8
β4 357 = 11 5 + β8 7
β4 5 + 3(7) = β11 The solution is β1,1
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Quick Review
β’ What are the dimensions of a matrix?β’ How do you find the product of two matrices?β’ What is an augmented matrix?β’ What are the characteristics of a matrix in row-echelon form?β’ What are the three row operations used to obtain row-echelon form?β’ How do you find the inverse of a 2Γ2 matrix?β’ Can the inverse be found for all dimensions of a matrix?β’ How do you solve a system of equations using Gaussian elimination?β’ How do you solve a system using the inverse of a matrix?