Page 39 - 35Linear Algebra

2.1 Gaussian Elimination 39

Entries left of the divide carry two indices; subscripts denote column number
and superscripts row number. We emphasize, the superscripts here do not
denote exponents. Make sure you can write out the system of equations and
the associated matrix equation for any augmented matrix.

Reading homework: problem 1

We now have three ways of writing the same question. Let’s put them
side by side as we solve the system by strategically adding and subtracting
equations. We will not tell you the motivation for this particular series of
steps yet, but let you develop some intuition first.

Example 11 (How matrix equations and augmented matrices change in elimination)

x + y = 27 1 1 x 27 1 1 27
⇔ = ⇔ .
2x − y = 0 2 −1 y 0 2 −1 0
With the first equation replaced by the sum of the two equations this becomes

3x + 0 = 27 3 0 x 27 3 0 27
⇔ = ⇔ .
2x − y = 0 2 −1 y 0 2 −1 0
Let the new first equation be the old first equation divided by 3:

x + 0 = 9 1 0 x 9 1 0 9
⇔ = ⇔ .
2x − y = 0 2 −1 y 0 2 −1 0
Replace the second equation by the second equation minus two times the first equation:

x + 0 = 9 1 0 x 9 1 0 9
⇔ = ⇔ .
0 − y = −18 0 −1 y −18 0 −1 −18
Let the new second equation be the old second equation divided by -1:

x + 0 = 9 1 0 x 9 1 0 9
⇔ = ⇔ .
0 + y = 18 0 1 y 18 0 1 18
Did you see what the strategy was? To eliminate y from the first equation
and then eliminate x from the second. The result was the solution to the
system.
Here is the big idea: Everywhere in the instructions above we can replace
the word “equation” with the word “row” and interpret them as telling us
what to do with the augmented matrix instead of the system of equations.
Performed systemically, the result is the Gaussian elimination algorithm.

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