Download matrices and systems of equations

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Capelli's identity wikipedia, lookup

Cartesian tensor wikipedia, lookup

History of algebra wikipedia, lookup

Quadratic form wikipedia, lookup

Equation wikipedia, lookup

Linear algebra wikipedia, lookup

Eigenvalues and eigenvectors wikipedia, lookup

Jordan normal form wikipedia, lookup

Symmetry in quantum mechanics wikipedia, lookup

Determinant wikipedia, lookup

Four-vector wikipedia, lookup

Singular-value decomposition wikipedia, lookup

Matrix (mathematics) wikipedia, lookup

Non-negative matrix factorization wikipedia, lookup

Perron–Frobenius theorem wikipedia, lookup

Matrix calculus wikipedia, lookup

System of linear equations wikipedia, lookup

Cayley–Hamilton theorem wikipedia, lookup

Matrix multiplication wikipedia, lookup

Transcript
8.1
MATRICES AND SYSTEMS OF EQUATIONS
Copyright © Cengage Learning. All rights reserved.
What You Should Learn
• Write matrices and identify their orders.
• Perform elementary row operations on matrices.
• Use matrices and Gaussian elimination to solve
systems of linear equations.
• Use matrices and Gauss-Jordan elimination to
solve systems of linear equations.
2
Matrices
3
Matrices
A matrix having m rows and n columns is said to be of order m  n.
If m = n, the matrix is square of order m  m (or n  n). For a square
matrix, the entries a11, a22, a33, . . . are the main diagonal entries.
4
Example 1 – Order of Matrices
Determine the order of each matrix.
a.
b.
Solution:
a. This matrix has one row and one column. The order of
the matrix is 1  1.
b. This matrix has one row and four columns. The order of
the matrix is 1  4.
5
Example 1 – Order of Matrices
Determine the order of each matrix.
c.
d.
Solution:
c. This matrix has two rows and two columns. The order of
the matrix is 2  2.
d. This matrix has three rows and two columns. The order
of the matrix is 3  2.
6
Matrices
Row matrix: A matrix that has only one row
Column matrix: Amatrix that has only one column
Augmented matrix of the system: A matrix derived from a
system of linear equations (each written in standard form
with the constant term on the right)
Coefficient matrix of the system: the matrix derived from
the coefficients of the system (but not including the
constant terms)
7
Matrices
System:
x – 4y + 3z = 5
–x + 3y – z = –3
2x
– 4z = 6
Augmented Matrix:
Coefficient Matrix:
8
Elementary Row Operations
9
Elementary Row Operations
1. Interchange two equations.
Interchange two rows
2. Multiply an equation by a nonzero constant.
Multiply a row by a nonzero constant
3. Add a multiple of an equation to another equation.
Add a multiple of a row to another row
10
Elementary Row Operations
1
0

0


0

0
1
0 1
 
0 0
0 0







 1 

 0 1
1
0

0


0

0
0
1
0

0
0
0
0
1

0
0






0
0
0

1
0
0
0
0


0

1
11
Gaussian Elimination with
Back-Substitution
12
Example 6 – Gaussian Elimination with Back-Substitution
Solve the system
y + z – 2w = –3
x + 2y – z
= 2 .
2x + 4y + z – 3w = – 2
x – 4y – 7z – w = –19
Solution:
Write augmented matrix.
13
Example 6 – Solution
cont’d
Interchange R1 and R2 so
first column has leading 1
in upper left corner.
Perform operations on R3
and R4 so first column has
zeros below its leading 1.
14
Example 6 – Solution
cont’d
Perform operations on R4
so second column has
zeros below its leading 1.
Perform operations on R3
and R4 so third and fourth
columns have leading 1’s.
15
Example 6 – Solution
cont’d
The matrix is now in row-echelon form, and the
corresponding system is
x + 2y – z
= 2
y + z – 2w = –3 .
z – w = –2
w= 3
Using back-substitution, you can determine that the
solution is x = –1, y = 2, z = 1, and w = 3.
16
Gaussian Elimination with Back-Substitution
17
Gauss-Jordan Elimination
18
Gauss-Jordan Elimination
With Gaussian elimination, elementary row operations are
applied to a matrix to obtain a (row-equivalent) row-echelon
form of the matrix.
A second method of elimination, called Gauss-Jordan
elimination, continues the reduction process until a
reduced row-echelon form is obtained.
19
Example 8 – Gauss-Jordan Elimination
Use Gauss-Jordan elimination to solve the system
x – 2y + 3z = 9
–x + 3y
= –4 .
2x – 5y + 5z = 17
Solution:
The row-echelon form of the linear system can be obtained
using the Gaussian elimination.
20
Example 8 – Solution
cont’d
Now, apply elementary row operations until you obtain
zeros above each of the leading 1’s, as follows.
Perform operations on R1
so second column has a
zero above its leading 1.
Perform operations on R1
and R2 so third column has
zeros above its leading 1.
21
Example 8 – Solution
cont’d
The matrix is now in reduced row-echelon form. Converting
back to a system of linear equations, you have
x=1
y = –1.
z=2
Now you can simply read the solution, x = 1, y = –1, and
z = 2 which can be written as the ordered triple(1, –1, 2).
22
Gaussian Elimination with
Back substitution
1
0

0


0

0
1
0 1
 
0 0
0 0







 1 

 0 1
Gaussian Jordon
1
0

0


0

0
0
1
0

0
0
0
0
1

0
0






0
0
0

1
0
0
0
0


0

1
23