How are column operations used to obtain Q in elementary column operations?

[QuantumP7]

I'm reading Charles G. Cullens' "Matrices and Linear Transformations" and have a question about one of the examples. Don't worry- I'm self-studying this.

Homework Statement

This is the section about matrix equivalence- both row and column equivalence. It says "Theorem 1.30 If a sequence of elementary operations on A applied to:

$$\begin{bmatrix} A & I \\ I & 0 \end{bmatrix}$$ yields

$$\begin{bmatrix} B & P \\ Q & 0 \end{bmatrix}$$ then PAQ = B." This example works with this theorem.

So, it has an example A = $$\begin{bmatrix} 1 & 2 & -1 & 2 \\ -2 & -5 & 3 & 0 \\ 1 & 0 & 1 & 10 \end{bmatrix}$$

with A being augmented: $$\begin{bmatrix} 1 & 2 & -1 & 2 & 1 & 0 & 0 \\ -2 & -5 & 3 & 0 & 0 & 1 & 0\\ 1 & 0 & 1 & 10 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & & & \\ 0 & 1 & 0 & 0 & & &\\ 0 & 0 & 1 & 0 & & &\\ 0 & 0 & 0 & 1 & & &\end{bmatrix}$$

Where the 3 x 3 identity matrix is for P and the 4 x 4 matrix is for Q. I understand the elementary row operations to obtain P, which turns out to be:
$$\begin{bmatrix} 1 & 0 & 0 \\ -2 & -1 & 0 \\ -5 & -2 & 1 \end{bmatrix}$$
. This is just transforming A and P into a row-reduced echelon form.

What I do not understand is the algorithm for obtaining Q (which turns out to be:

$$\begin{bmatrix} 1 & -2 & -1 & -10 \\ 0 & 1 & 1 & 4 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$ )

How is this Q obtained? Are we trying to reduce the matrix for Q to a column reduced echelon form? If so, what exactly is column reduced echelon form? The book lists steps for obtaining Q, but does not explain them. Here are the steps:
(-2$$C_{1} + C_{2}$$)
($$C_{1} + C_{3}$$)
(-2$$C_{1} + C_{4}$$)

then

($$C_{2} + C_{3}$$)
(4$$C_{2} + C_4}$$)

Homework Equations

The Attempt at a Solution

 

[lippyka]

I have read the book and attempted to apply the steps to the matrix, but I do not understand what is going on. I know that row/column equivalence has to do with elementary row/column operations, but I do not understand how these operations yield Q.