If matrix \(A_p =\begin{bmatrix}p&(p+1)\\p&(p-1)\end{bmatrix}, p \in N\) (where N is the set of natural numbers), then the value of \(|A_1|+|A_2|+|A_3|+...+|A_{2025}|\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(-(2025) (2026)\)
Each determinant is
\(|A_p| = p(p - 1) - p(p + 1)\)
\(= p^2 - p - p^2 - p\)
\(= -2p\)
Adding over \(p = 1\) to 2025 and using the sum of the first n natural numbers, \(\frac{n(n+1)}{2}\):
\( \displaystyle \sum_{p=1}^{2025} |A_p| = -2\sum_{p=1}^{2025} p \)
\( \displaystyle = -2 \times \frac{2025 \times 2026}{2} \)
\(= -(2025)(2026)\)