Working on pivots intuition

Question

I am learning linear algebra using Gilbert Strangs "Intro to LA" 4th edition. On problem set for chapter 2.3 "Elimination Using Matrices" I encountered a question I can't wrap my head around(problem 5):

Suppose a₃₃ = 7 and the third pivot is 5. If you change a₃₃ to 11, the third pivot is ____. If you change a₃₃ to ____, there is no third pivot.

Problem 3 mentions matrix A:

A = [[1 1 0]
     [4 6 1]
     [-2 2 0]]

I am assuming problem 5 refers to that matrix. The answer is 9 and 2 respectively. I don't understand this question at all. If a₃₃ is the coefficient matrix then a₃₃ = 7 means the third is pivot is 7, how can it be 5 at the same time. I feel like I am missing some fundamental concept there...

Answer 1

The $k^\text{th}$ pivot, if I remember things from that book correctly, is defined as the first nonzero entry in the $k^\text{th}$ row, after row-reduction (ref form). So the pivots would only necessarily correspond to the diagonal entries if the matrix is upper-triangular (with nonzero entries on the diagonal).