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Get Full Access to La Salle - MTH 24001 - Study Guide - Midterm
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LA SALLE / OTHER / MTH 24001 / What is augmented matrix?

# What is augmented matrix? Description

##### Description: Study guide for test 2 in MTH 240-01 with Professor Andrilli covering section 2.1-3.3 of the textbook
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Math 240 test number 2 - La Salle UniversityDon't forget about the age old question of Who are the key resources in business?

Section 2.1: Systems of linear equations

3 Row Operations: Gaussian Elimination

1. Multiply any row by a nonzero number to create pivots
2. Add any multiple of one row to another row to create zeros
3. Switch any two rows

Augmented matrix [A|B]If you want to learn more check out What is the meaning of strategos?

Coefficient matrix Ax = BIf you want to learn more check out What kinds of issues can elasticity help us understand?

Number of solutions to a system:

1. A single solution↘
2. Infinite solutions → consistent system
3. No solutions - inconsistent

If a column is not a pivot, its variable ERDon't forget about the age old question of What are the 4 major fields of anthropology?

Dependent variables = pivotsIf you want to learn more check out What does the proportion of income mean?

independent variables = no pivots

For inconsistent systems, the final augmented matrix always contains at least one row:We also discuss several other topics like What do you call the isomers with the same formula but different connectivity of atoms?

[0 0,.. 0 | C] C  0

Of c = 0, ignore row and solve and solve system using back substitution

Solution set = {(        ) | C E R}

Number 11         (a)T        (c)F        (e)T        (g)F

(b)F        (d)F        (f)T

Section 2.2: Reduced Row Echelon Form

Row Echelon Form = staircase pattern of pivots

Gauss - Jordan Method

• Zero out entries above each pivot as well
• Fewer non zero numbers in final augmented matrix so solution set is more apparent

Reduced Row Echelon Form

1. 1rst non zero number in each row is a 1
2. Each successive row has its 1rst non zero number in a later column
3. All entries above and below pivots are zero
4. All full rows of zeros are in final rows

Homogeneous System: AX = 0 ← zero matrix

• Augmented matrix [A | 0]
• Never inconsistent, always consistent
• Has 1 solution (trivial solutions) or infinite solutions (non trivial)

CORR 2.3 if a system of “m” equations and “n” unknowns is homogeneous and m < n, then there is an infinite number of solutions

• “Zero” solutions never occurs

Application: balancing chemical equations

• Empirical formula → coulis each element and created an augmented matrix solving several systems simultaneously
• Augmented matrix [A | BC]

Number 14        (a)T        (c)F        (e)F

(b)T        (d)T        (f)F

Section 2.3: Equivalent Systems

Two systems of “m” linear equations in “n” variables are equivalent if they have exactly the same solution set

A matrix C is row equivalent to a matrix D if D is obtained from C by a finite number of row operations, type (I), (II), and (III)

• Every operation has a reversal
1. <i> ← c <i>                         <i> ← <i>
2. <i> ← c <i> + <j>                 <j> ← -c<i> + <j>
3. <i> ↔ <i>                        <i> ↔ <j>

THM 2.4 (RE) = row equivalent

Let C,D,E be matrices of same size

1. C (RE) D, so D (RE) C
2. If C (RE) D and D (RE) E, then C (RE) E

THM 2.5 if Ax = B is a system of linear equations and if [A | B] is row equivalent to [C,D] then Cx = D is equivalent to Ax = B

• Some solutions; justification for rref

THM 2.6 Every matrix is row equivalent to a unique rref matrix

Rank of matrix = number of pivot in the rref of the matrix

THM 2.7

If rank (A) < n = infinite solutions

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