Solve the following math problem efficiently and clearly:

- For simple problems (2 steps or fewer):
Provide a concise solution with minimal explanation.

- For complex problems (3 steps or more):
Use this step-by-step format:

## Step 1: [Concise description]
[Brief explanation and calculations]

## Step 2: [Concise description]
[Brief explanation and calculations]

...

Regardless of the approach, always conclude with:

Therefore, the final answer is: $oxed{answer}$. I hope it is correct.

Where [answer] is just the final number or expression that solves the problem.

Problem: What is the matrix $\mathbf{M}$ that performs the transformation which sends square $ABCD$ to square $A'B'C'D'$?  (In particular, $A$ goes to $A',$ and so on.)

[asy]

size(200);
import graph;

pair Z=(0,0), A=(2,3), B=(-3,2), C=(-4,1), D=(-1,-4);

Label f; 
f.p=fontsize(6); 
xaxis(-1.5,1.5,Ticks(f, 1.0)); 
yaxis(-0.5,2.5,Ticks(f, 1.0));

draw((0,0)--(1,0)--(1,1)--(0,1)--(0,0),red);
dot((0,0)); label("$A=A'$", (0,0), SE);
dot((1,0)); label("$D$", (1,0), NE);
dot((1,1)); label("$C=D'$", (1,1), E);
dot((0,1)); label("$B$", (0,1), NE);

draw((0,0)--(1,1)--(0,2)--(-1,1)--(0,0), blue);
dot((0,2)); label("$C'$", (0,2), NE);
dot((-1,1)); label("$B'$", (-1,1), W);

[/asy]
user