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Lecture 10

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University of Waterloo

Mathematics

MATH 136

Robert Sproule

Winter

Description

Monday, January 27 − Lecture 10 : Gauss-Jordan elimination algorithm for solving
systems of linear equations.
Concepts:
1. Gauss-Jordan elimination algorithm for solving systems of linear equations.
2. consistent and inconsistent system.
3. ERO’s
We see that solving a system of linear equations which is in RREF is a relatively easy
task. However most systems we encounter are not in RREF. We now look at a technique
where we transform such systems in RREF without changing the solution.
10.1 Definition − The Gauss-Jordan elimination is an algorithm which transforms a
system of linear equations onto an RREF system in such a way that the resulting system
has the same solution as the original system of equations.
These changes are done by performing 3 "Elementary row operations" (ERO’s) on the
augmented matrix associated to the given system :
- ERO of type I : Interchange two rows of the system. We denote this row operation by
P ijmeaning we "permute" or exchange the i and j rows. th
- ERO of type II : Multiply a row of the system by a non-zero real number. We denote
this by cR, mianing we multiply the row i with the scalar c.
- ERO of type III : Replace the i row R with the sum of R and the scalar multiple of
i i
row R j. We denote this by R + cR. i j
Note: We could actually apply the above operations to the system of equations and
referring to these as the 3 “equation-operations”:
- “ P ij, interchange the order of two equations,
- “cE”,imultiply both sides of equation E by a non-zeio scalar c,
- “E +icE”, rejlace equation E, by the sui of the equation E and the equation i
obtained by multiply both sides of the equation E jby c.
Anyone of these 3 operations does not change the solution of the system.
The elementary row operations essentially describe the operations performed above in
solving the system of linear equations. 10.1.1 Note − Performing the elementary operations on a system always results in a
system which has the same solution as the system we started off with.
10.1.2 Remark − Any ERO can be "undone" by an ERO:
- To "undo" P apijy P ij
- To "undo" cR appiy (1/c)R. i
- To "undo" R + ci applyjR + (−c)Ri j.
Verify this with a simple matrix.
10.1.3 Remark − For some there may be some confusion about the interpretation
given to an ERO such as, for example, 2R + R . 3 5
- The ERO “2R + R 3 is t5 be interpreted in the same ways as “R + 2R ”. It 5 3
means
“Replace row 5 with 2 times row 3”
- Just to avoid confusion some may prefer to state the ERO as:
R → R + 2R
5 5 3
which more clearly states “replace row 5 with 2 times row 3”.
- But do not write “R → 23 + R ”,3which5will surely confuse the reader.
10.2 General strategy for solving linear systems by using the Gauss-Jordan elimination.
System of linear equations
→ Obtain the augmented coefficient matrix [A|b]
→ Apply ERO’s to transform this matrix matrix [A|b] in RREF, [A RREF | d]
→ Transform back into the system of linear equations of [A RREF| d]
→ Isolate the basic variables.
→ Write out the solution set in scalar form.
→ Write out the solution in “vector equation form”.
10.2.1 Note − If the operations are done accurately the solution to the system of linear
equation [A RREF | d] is identical

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