The correct answer is $\boxed{\left[ {\begin{array}{*{20}{c}} { – 1}&0 \ 0&1 \end{array}} \right]}$.
A coordinate transformation matrix is a matrix that maps points in one coordinate system to points in another coordinate system. In this case, we are interested in the coordinate transformation matrix that maps points in the original coordinate system (where shape ABC is located) to points in the reflected coordinate system (where shape A1B1C1 is located).
The reflection across the horizontal axis (X-axis) is a linear transformation, which means that it can be represented by a matrix. The matrix that represents this transformation is a diagonal matrix with $-1$ on the diagonal. This is because the reflection across the horizontal axis maps each point $(x, y)$ to the point $(x, -y)$.
Therefore, the coordinate transformation matrix that maps ABC to A1B1C1 is $\boxed{\left[ {\begin{array}{*{20}{c}} { – 1}&0 \ 0&1 \end{array}} \right]}$.
Here is a brief explanation of each option:
- Option A: This matrix maps each point $(x, y)$ to the point $(x, y)$. This is not a reflection across the horizontal axis.
- Option B: This matrix maps each point $(x, y)$ to the point $(-x, y)$. This is a reflection across the vertical axis.
- Option C: This matrix maps each point $(x, y)$ to the point $(x, -y)$. This is the correct answer.
- Option D: This matrix maps each point $(x, y)$ to the point $(-x, -y)$. This is a reflection across both the horizontal and vertical axes.