We are given two matrices, $P$ and $Q$, and asked to find the result of $P - Q$.

Matrix $P$ is: $P = \begin{bmatrix} 1 & -2 \\ 3 & 4 \end{bmatrix}$

Matrix $Q$ is: $Q = \begin{bmatrix} 3 & -1 \\ 2 & -7 \end{bmatrix}$

Now, subtract matrix $Q$ from matrix $P$ element-wise:

$P - Q = \begin{bmatrix} (1 - 3) & (-2 - (-1)) \\ (3 - 2) & (4 - (-7)) \end{bmatrix}$

Perform the calculations:

$P - Q = \begin{bmatrix} -2 & -1 \\ 1 & 11 \end{bmatrix}$

Thus, the result of $P - Q$ is: $\begin{bmatrix} -2 & -1 \\ 1 & 11 \end{bmatrix}$

This corresponds to option C.

Would you like further details or have any questions? Here are some related questions:

1. What is the definition of matrix subtraction?
2. How does matrix addition work in comparison to subtraction?
3. Can you multiply matrices element-wise like addition and subtraction?
4. What properties of matrices are important when performing subtraction?
5. How do you calculate the determinant of a matrix?

Tip: Always ensure that the matrices involved in addition or subtraction are of the same dimensions.