A dilation with scale factor 7 multiplies each coordinate by 7.
Thus, the matrix $\mathbf{D}$ is given by $\mathbf{D} = \begin{pmatrix} 7 & 0 \\ 0 & 7 \end{pmatrix}$.
The determinant of a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is given by $ad - bc$.
Therefore, $\det \mathbf{D} = 7 \cdot 7 - 0 \cdot 0 = \boxed{49}$.The answer is: 49
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
Let $\mathbf{D}$ be the $2 \times 2$ matrix corresponding to the dilation, centered at the origin, with scale factor 7.  Find $\det \mathbf{D}.$

### Answer:
