Strang, Intro. to Linear Algebra, Section 2.3, Problem 29: what does it mean that Pascal's matrix is exceptional since all of its multipliers are 1. Consider the matrix
$$\begin{bmatrix}
1 & 0 & 0 & 0\\
1 & 1 & 0 & 0\\
1 & 2 & 1 & 0\\
1 & 3 & 3 & 1
\end{bmatrix}$$
In Strang's Introduction to Linear Algebra, 5th Edition Section 2.3 Problem 29 he says that this is Pascal's Matrix and that

Pascal's triangular matrix is exceptional, all of its multipliers
$l_{ij}=1$.

Why is this matrix exceptional? Also, it seems the multipliers required in forward elimination for this matrix do not seem to all be 1. Is that a typo or am I missing something?
 A: It's probably just misworded.
The book's claim is true in the following not-really-valid sense: You can fully row-reduce this matrix using $1$ as your multiplier every time, but only if you make very specific and nonstandard choices for the order in which to eliminate entries and also for which row to use for eliminating certain entries. You'd need to do the following steps in this order:

*

*Subtract row 3 from row 4

*Subtract row 2 from row 3

*Subtract row 1 from row 2

At this point you've reached the intermediate point $\left(\begin{array}{cccc} 
1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 1 & 2 & 1 \end{array}\right)$.


*Subtract row 3 from row 4.

*Subtract row 2 from row 3.

*Subtract row 3 from row 4.

...and at this point you've created the identity matrix. This order basically comes down to integrating the subdiagonal entries in the first column from bottom to top, then handling the second column from bottom to top, and so on.
That said, if you just generically refer to "the multipliers $l_ij$ that arise during Gaussian elimination for matrix $M$" with no extra information, then absolutely everyone will assume you mean the "standard" order, where we eliminate each column's subdiagonal entries from top to bottom using the diagonal "pivot" entry from that column. So you're correct to be confused by the book's statement.
If you a) understand how it's possible to reduce that particular matrix while only using multipliers of $1$ by choosing your operations in a weird way, but also b) understand that the wording here is imprecise/wrong and Gaussian elimination does have a certain standard ordering, then I think you've gotten all the learning you're going to get from this exercise and I would just not worry about it further.
