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Solved Problems on Gauss Jordan Elimination | MATH 1630, Exams of Mathematics

Material Type: Exam; Class: MATH 1630: If high school precalculus and ACT math of at least 21 contact 694-6450.; Subject: Mathematics; University: Pellissippi State Technical Community College; Term: Unknown 1989;

Typology: Exams

Pre 2010

Uploaded on 08/17/2009

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Gauss-Jordan - page 1
Gauss-Jordan Elimination
I. A linear system such as can be solved by hand algebraically or by using an
x5yz11
3z12
2x4y2z8
+=−
=
+−=
augmented matrix and elementary row operations.
A. Elementary row operations that produce row-equivalent matrices
1. Two rows are interchanged
ij
RR
2. A row is multiplied by a nonzero constant
ii
kRR
3. A constant multiple of one row is added to another row
jii
kRRR
+→
(NOTE:means"replaces")
II. Performing elementary row operations on the TI-83
The result of a row operation is displayed on the home screen, but it is not automatically stored!
You should immediately store the result under a different name. It is convenient (and frequently
useful) to store the results alphabetically.
A. Row swap
To interchange rows 1 and 3 of matrix A:
MATH C:rowSwap( MATRIX ENTER MATRIX
NAMES 1:[A]
ENTER,1,3)
NAMES 2:[B]
ENTER STO$MATRIX ENTER
B. Multiplying a row by a nonzero scalar
To multiply row 1 of matrix A by :
1
3
MATH E:*row(
MATRIX ENTER 1
3, MATRIX
NAMES 1:[A]
ENTER , 1 ) ENTER
NAMES 3:[C] STO$MATRIX ENTER
C. Adding a nonzero scalar multiple of one row to another row
To multiply row 2 of matrix A by and add it to row 3
1
2
(of matrix A):
MATH F:*row+(
MATRIX ENTER 1
2
, MATRIX
NAMES 1:[A]
ENTER , 2 , 3 ) ENTER
NAMES 4:[D]
STO$MATRIX ENTER
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Gauss-Jordan Elimination

I.

A linear system such as can be solved by hand algebraically or by using an

x 5y z 11

3z 12

2x 4y 2z 8

augmented matrix and elementary row operations.

A. Elementary row operations that produce row-equivalent matrices

  1. Two rows are interchanged

i j

R ↔R

  1. A row is multiplied by a nonzero constant

i i

kR →R

  1. A constant multiple of one row is added to another row

j i i

kR + R →R

(NOTE : →means "replaces")

II. Performing elementary row operations on the TI-

The result of a row operation is displayed on the home screen, but it is not automatically stored!

You should immediately store the result under a different name. It is convenient (and frequently

useful) to store the results alphabetically.

A. Row swap

To interchange rows 1 and 3 of matrix A:

MATRIX MATH C:rowSwap( ENTER MATRIX

NAMES 1:[A]

ENTER , 1 , 3 )

NAMES 2:[B]

ENTER STO∃ MATRIX ENTER

B. Multiplying a row by a nonzero scalar

To multiply row 1 of matrix A by :

MATH E:*row( MATRIX ENTER

, MATRIX

NAMES 1:[A]

ENTER , 1 ) ENTER

STO∃ MATRIX NAMES 3:[C] ENTER

C. Adding a nonzero scalar multiple of one row to another row

To multiply row 2 of matrix A by and add it to row 3

(of matrix A):

MATH F:*row+(

MATRIX ENTER

, MATRIX

NAMES 1:[A]

ENTER , 2 , 3 ) ENTER

NAMES 4:[D]

STO∃ MATRIX ENTER

III. Solve the system of equations represented by the given augmented matrix using the given row

operations:

As you perform each row operation, record the result, and store it as indicated.

Let [A] =

matrix operation result store as matrix

A. [B]

1 3

R ↔R

B. [C]

1 2 2

−2R + R →R

C. [D]

2 2

R R

D. [E]

2 1 1

−5R + R →R

E. [F]

3 3

R R

F. [G]

3 1 1

R + R →R

The solution (x, y, z) should be in the column to the right of the bar. (x, y, z) = _____________

IV. This system could have been solved using different row operations and/or the same row operations

in other orders. To minimize the amount of work necessary to solve the system, you must be

careful not to backtrack and “undo” work which you have already done. The process we have been