If matrix $A_p =\begin{bmatrix}p&(p+1)\\p&(p-1)\end{bmatrix},p∈N$ (where N is the set of natural numbers), then the value of $|A_1|+|A_2|+|A_3|+...+|A_{2025}|$ is

$-(2025) (2026)$

The correct answer is Option (1) → $-(2025) (2026)$ **

Given $A_p=\begin{pmatrix}p & p+1\\[4pt] p & p-1\end{pmatrix}$.

Determinant: $|A_p|=p(p-1)-p(p+1)=p\big((p-1)-(p+1)\big)=p(-2)=-2p$.

Sum: $\displaystyle \sum_{p=1}^{2025}|A_p|=\sum_{p=1}^{2025}(-2p)=-2\sum_{p=1}^{2025}p =-2\cdot\frac{2025\cdot2026}{2}$.

$\displaystyle -\,2025\cdot2026$