Step 1: Compute $A^{2}$

Given $A = \begin{bmatrix}-p & q \\ r & p\end{bmatrix}$. We calculate $A^{2} = A \times A$:

$$A^{2} = \begin{bmatrix}-p & q \\ r & p\end{bmatrix} \begin{bmatrix}-p & q \\ r & p\end{bmatrix}$$ $$A^{2} = \begin{bmatrix}(-p)(-p) + (q)(r) & (-p)(q) + (q)(p) \\ (r)(-p) + (p)(r) & (r)(q) + (p)(p)\end{bmatrix}$$ $$A^{2} = \begin{bmatrix}p^{2} + qr & -pq + pq \\ -pr + pr & qr + p^{2}\end{bmatrix} = \begin{bmatrix}p^{2} + qr & 0 \\ 0 & p^{2} + qr\end{bmatrix}$$

Step 2: Equate to Identity Matrix $I$

We are given $A^{2} = I$, where $I = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$. Therefore:

$$\begin{bmatrix}p^{2} + qr & 0 \\ 0 & p^{2} + qr\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$$

Step 3: Solve for the relation

By comparing the corresponding elements, we get:

$$p^{2} + qr = 1$$ $$p^{2} + qr - 1 = 0$$

Multiplying by -1 to match the options:

$$1 - p^{2} - qr = 0$$