SciTech-Math-AdvancedAlgebra- Cramer' Rule (Gabriel Cramer (1704–1752)) + Gauss-Jordan Method

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METHODLOGY: Qualitative/Quantitative, Static/Dynamic

APPROACH: Advanced Algebra,

• SYSTEM: systems of Linear Equations -> Matrix:{Coefficient Matrix, Augmented Matrix},
• STRATEGY: Elimination Method, Gauss-Jordan method
• TACTIC: Simple Row Operations, Tasks(arithmetic manipulation):
  • multipliedByANonZeroConstant,
  • aConstantMultipleOfArowAddedToAnotherRow
  • InterchangeTwoRows
    
    MILESTONE: Row Echelon Form( The reduced row echelon form of the coefficient matrix)
  • has 1's along the main diagonal and zeros elsewhere
    
    The solution is readily obtained from this form.

K: Domain K

K^n: N-Dim Vector Space:

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VOCABULARY

ech·e·lon, | ˈeSHəˌlän |,

noun

1. a level or rank in an organization, a profession, or society: the upper echelons of the business world.
2. Military a formation of troops, ships, aircraft, or vehicles in parallel rows with the end of each row projecting further than the one in front: the regiment lined up shoulder to shoulder in three tight echelons | [mass noun] : there are two planes, lying in echelon with one another.
   
   • [often with modifier] a part of a military force differentiated by position in battle or by function: the rear echelon.

verb [with object] Military

arrange in an echelon formation: (as noun echeloning) : the echeloning of fire teams.

ORIGIN

late 18th century (in echelon (sense 2 of the noun)): from French échelon, from échelle ‘ladder’, from Latin scala.

2.2: Systems of Linear Equations and the Gauss-Jordan Method

Learning Objectives

In this section you will learn to

• Represent a system of linear equations as an augmented matrix
• Solve the system using elementary row operations.

In this section, we learn to solve systems of linear equations using a process called the Gauss-Jordan method.

The process:

1. begins by first expressing the system as a matrix,
   
   we expressed the system of equations as = , where:

   • represented the coefficient matrix,
   • the matrix of constant terms.
   • [] As an augmented matrix, we write the matrix as [] .
   • It is clear that all of the information is maintained in this matrix form,
     
     and only the letters , and are missing. A student may choose to write , and on top of the first three columns to help ease the transition.

2. and then reducing it to an equivalent system by simple row operations.
   
   Now we list the three row operations the Gauss-Jordan method employs. Row Operations:

   • Any two rows in the augmented matrix may be interchanged.
   • Any row may be multiplied by a non-zero constant.
   • A constant multiple of a row may be added to another row.

3. The process is continued until the solution is obvious from the matrix.
   
   The reduced row echelon form of the coefficient matrix:
   
   has 1's along the main diagonal and zeros elsewhere.
   
   The solution is readily obtained from this form.

The matrix that represents the system is called the augmented matrix,

and the arithmetic manipulation that is used to move,

from a system to a reduced equivalent system is called a row operation.

Once a system is expressed as an augmented matrix,

the Gauss-Jordan method reduces the system,

into a series of equivalent systems by using the row operations.

This row reduction continues until the system is expressed in what is called the reduced row echelon form.

The method is not much different form the algebraic operations we employed in the elimination method in the first chapter. The basic difference is that it is algorithmic in nature, and, therefore, can easily be programmed on a computer.

We will next solve a system of two equations with two unknowns, using the elimination method, and then show that the method is analogous to the Gauss-Jordan method.

Gauss-Jordan Method

• Write the augmented matrix.
• Interchange rows if necessary to obtain a non-zero number in the first row, first column.
• Use a row operation to get a 1 as the entry in the first row and first column.
  
  Use row operations to make all other entries as zeros in column one.
• Interchange rows if necessary to obtain a nonzero number in the second row, second column.
  
  Use a row operation to make this entry 1. Use row operations to make all other entries as zeros in column two.
• Repeat step 5 for row 3, column 3.
• Continue moving along the main diagonal until you reach the last row, or until the number is zero.
• The final matrix is called the reduced row-echelon form.

[9.8: Solving Systems with Cramer's Rule]

Learning Objectives

• Evaluate 2 × 2 determinants.: Use Cramer’s Rule to solve a system of equations in two variables.
• Evaluate 3 × 3 determinants.: Use Cramer’s Rule to solve a system of three equations in three variables.
• Know the properties of determinants.
  
  PROPERTIES OF DETERMINANTS
  • If the matrix is in upper triangular form, the determinant equals the product of entries down the main diagonal.
  • When two rows are interchanged, the determinant changes sign.
  • If either two rows or two columns are identical, the determinant equals zero.
  • If a matrix contains either a row of zeros or a column of zeros, the determinant equals zero.
  • The determinant of an inverse matrix −1 is the reciprocal of the determinant of the matrix .
  • If any row or column is multiplied by a constant, the determinant is multiplied by the same factor.