Matrix Equations and Systems of Linear Equations

Сентябрь 13, 2022

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Matrix Equations and Systems of Linear Equations

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2. Barnett/Ziegler/Byleen Finite Mathematics 11e Matrix Equations Let’s review one property of solving equations involving real numbers.
3. Barnett/Ziegler/Byleen Finite Mathematics 11e Basic Properties of Matrices Assuming that all products and sums are defined
4. Barnett/Ziegler/Byleen Finite Mathematics 11e Basic Properties of Matrices (continued) Multiplication Properties Associative Property: A(BC) = (AB)C
5. Barnett/Ziegler/Byleen Finite Mathematics 11e Basic Properties of Matrices (continued) Equality Addition: If A = B, then
6. Barnett/Ziegler/Byleen Finite Mathematics 11e Solving a Matrix Equation Reasons for each step: Given; since A is
7. Barnett/Ziegler/Byleen Finite Mathematics 11e Example Example: Use matrix inverses to solve the system
8. Barnett/Ziegler/Byleen Finite Mathematics 11e Example Example: Use matrix inverses to solve the system Solution: Write out
9. Barnett/Ziegler/Byleen Finite Mathematics 11e Example (continued) Form the matrix equation AX = B. Multiply the 3
10. Barnett/Ziegler/Byleen Finite Mathematics 11e Example (continued) If the matrix A-1 exists, then the solution is determined
11. Barnett/Ziegler/Byleen Finite Mathematics 11e Example Solution The product of A-1 and B is The solution can
12. Barnett/Ziegler/Byleen Finite Mathematics 11e Another Example Example: Solve the system on the right using the inverse
13. Barnett/Ziegler/Byleen Finite Mathematics 11e Another Example Example: Solve the system on the right using the inverse
14. Barnett/Ziegler/Byleen Finite Mathematics 11e Cases When Matrix Techniques Do Not Work There are two cases when
15. Barnett/Ziegler/Byleen Finite Mathematics 11e Application Production scheduling: Labor and material costs for manufacturing two guitar models
16. Barnett/Ziegler/Byleen Finite Mathematics 11e Solution Let x be the number of model A guitars to produce
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Слайд 2

Barnett/Ziegler/Byleen Finite Mathematics 11e
Matrix Equations
Let’s review one property of solving equations

involving real numbers. Recall
If ax = b then x = , or
A similar property of matrices will be used to solve systems of linear equations.
Many of the basic properties of matrices are similar to the properties of real numbers, with the exception that matrix multiplication is not commutative.

Слайд 3

Barnett/Ziegler/Byleen Finite Mathematics 11e
Basic Properties of Matrices
Assuming that all products and

sums are defined for the indicated matrices A, B, C, I, and 0, we have
Addition Properties
Associative: (A + B) + C = A + (B+ C)
Commutative: A + B = B + A
Additive Identity: A + 0 = 0 + A = A
Additive Inverse: A + (-A) = (-A) + A = 0

Слайд 4

Barnett/Ziegler/Byleen Finite Mathematics 11e
Basic Properties of Matrices (continued)
Multiplication Properties
Associative Property: A(BC) =

(AB)C
Multiplicative identity: AI = IA = A
Multiplicative inverse: If A is a square matrix and A-1 exists, then AA-1 = A-1A = I
Combined Properties
Left distributive: A(B + C) = AB + AC
Right distributive: (B + C)A = BA + CA

Слайд 5

Barnett/Ziegler/Byleen Finite Mathematics 11e
Basic Properties of Matrices (continued)
Equality
Addition: If A = B,

then A + C = B + C
Left multiplication: If A = B, then CA = CB
Right multiplication: If A = B, then AC = BC

The use of these properties is best illustrated by an example of solving a matrix equation.
Example: Given an n x n matrix A and an n x p matrix B and a third matrix denoted by X, we will solve the matrix equation AX = B for X.

Слайд 6

Barnett/Ziegler/Byleen Finite Mathematics 11e
Solving a Matrix Equation
Reasons for each step:
Given; since

A is n x n, X must by n x p.
Multiply on the left by A-1.
Associative property of matrices
Property of matrix inverses.
Property of the identity matrix
Solution. Note A-1 is on the left of B. The order cannot be reversed because matrix multiplication is not commutative.

Слайд 7

Barnett/Ziegler/Byleen Finite Mathematics 11e
Example
Example: Use matrix inverses to solve the

system

Слайд 8

Barnett/Ziegler/Byleen Finite Mathematics 11e
Example
Example: Use matrix inverses to solve the

system
Solution:
Write out the matrix of coefficients A, the matrix X containing the variables x, y, and z, and the column matrix B containing the numbers on the right hand side of the equal sign.

Слайд 9

Barnett/Ziegler/Byleen Finite Mathematics 11e
Example (continued)
Form the matrix equation AX = B. Multiply

the 3 x 3 matrix A by the 3 x 1 matrix X to verify that this multiplication produces the 3 x 3 system at the bottom:

Слайд 10

Barnett/Ziegler/Byleen Finite Mathematics 11e
Example (continued)
If the matrix A-1 exists, then the solution

is determined by multiplying A-1 by the matrix B. Since A-1 is 3 x 3 and B is 3 x 1, the resulting product will have dimensions 3 x 1 and will store the values of x, y and z.
A-1 can be determined by the methods of a previous section or by using a computer or calculator. The resulting equation is shown at the right:

Слайд 11

Barnett/Ziegler/Byleen Finite Mathematics 11e
Example Solution
The product of A-1 and B is
The solution

can be read off from the X matrix: x = 0, y = 2, z = -1/2
Written as an ordered triple of numbers, the solution is (0, 2, -1/2)

Слайд 12

Barnett/Ziegler/Byleen Finite Mathematics 11e
Another Example
Example: Solve the system on the right

using the inverse matrix method.

Слайд 13

Barnett/Ziegler/Byleen Finite Mathematics 11e
Another Example
Example: Solve the system on the right

using the inverse matrix method.
Solution:
The coefficient matrix A is displayed at the right. The inverse of A does not exist. (We can determine this by using a calculator.) We cannot use the inverse matrix method. Whenever the inverse of a matrix does not exist, we say that the matrix is singular.

Слайд 14

Barnett/Ziegler/Byleen Finite Mathematics 11e
Cases When Matrix Techniques Do Not Work
There are

two cases when inverse methods will not work:
1. If the coefficient matrix is singular
2. If the number of variables is not the same as the number of equations.

Слайд 15

Barnett/Ziegler/Byleen Finite Mathematics 11e
Application
Production scheduling: Labor and material costs for

manufacturing two guitar models are given in the table below: Suppose that in a given week $1800 is used for labor and $1200 used for materials. How many of each model should be produced to use exactly each of these allocations?

Слайд 16

Barnett/Ziegler/Byleen Finite Mathematics 11e
Solution
Let x be the number of model A

guitars to produce and y represent the number of model B guitars. Then, multiplying the labor costs for each guitar by the number of guitars produced, we have
30x + 40y = 1800
Since the material costs are $20 and $30 for models A and B respectively, we have 20x + 30y = 1200.

This gives us the system of linear equations:
30x + 40y = 1800
20x + 30y = 1200
We can write this as a matrix equation:

Matrix Equations and Systems of Linear Equations

Содержание

Barnett/Ziegler/Byleen Finite Mathematics 11eMatrix EquationsLet’s review one property of solving equations

Barnett/Ziegler/Byleen Finite Mathematics 11eBasic Properties of MatricesAssuming that all products and

Barnett/Ziegler/Byleen Finite Mathematics 11eBasic Properties of Matrices (continued)Multiplication PropertiesAssociative Property: A(BC) =

Barnett/Ziegler/Byleen Finite Mathematics 11eBasic Properties of Matrices (continued)EqualityAddition: If A = B,

Barnett/Ziegler/Byleen Finite Mathematics 11eSolving a Matrix EquationReasons for each step:Given; since

Barnett/Ziegler/Byleen Finite Mathematics 11eExample Example: Use matrix inverses to solve the

Barnett/Ziegler/Byleen Finite Mathematics 11eExample Example: Use matrix inverses to solve the

Barnett/Ziegler/Byleen Finite Mathematics 11eExample (continued)Form the matrix equation AX = B. Multiply

Barnett/Ziegler/Byleen Finite Mathematics 11eExample (continued)If the matrix A-1 exists, then the solution

Barnett/Ziegler/Byleen Finite Mathematics 11eExample SolutionThe product of A-1 and B isThe solution

Barnett/Ziegler/Byleen Finite Mathematics 11eAnother ExampleExample: Solve the system on the right

Barnett/Ziegler/Byleen Finite Mathematics 11eAnother ExampleExample: Solve the system on the right

Barnett/Ziegler/Byleen Finite Mathematics 11eCases When Matrix Techniques Do Not WorkThere are

Barnett/Ziegler/Byleen Finite Mathematics 11eApplication Production scheduling: Labor and material costs for

Barnett/Ziegler/Byleen Finite Mathematics 11eSolutionLet x be the number of model A

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Example Solution
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Another Example
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Solution
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