What Changes in the Third Pivot When a3,3 Shifts from 7 to 11 in Linear Algebra?

[kostoglotov]

Course: MIT OCW 18.06 Intro to Linear Algebra by Strang 4th edt.

Question: if a_3,3 is 7, and the third pivot is 5, if we change a_3,3 to be 11, then the third pivot becomes _________. If you change the a_3,3 to ________ then there is no third pivot.

At first I thought a_3,3 had to be the actual third pivot (yes?no?). Definitions for what a pivot is are muddled throughout the resources I've used, but the most solid definition I've found is from the course text itself and it says (to paraphrase):

1 >> The pivots are on the diagonal of the upper triangular matrix after elimination is complete whereby back substitution may begin

To not paraphrase it also says

2 >> "pivot = 1st nonzero in the row that does the elimination"

From the first definition, shouldn't a_3,3 be the pivot?

If not, then perhaps there is a 5 leading the third row...ok. But then if we changed the 7 to an 11, that would be by multiplying all the elements of the row by a constant multiple, and the answer in the solutions guide says that changing a_3,3 from 7 to 11 changes the pivot from 5 to 9...

I am clearly lacking some fundamental understanding in what a pivot is. Can someone help?

 

[RUber]

It looks like when they say change $a_{3,3}$ from 7 to 11, they are not implying a multiplication, but a shift. (addition).
I am not fully versed on the notation and definitions you are using, but based on the example you gave, it looks like they are calling for addition.