If A_k=\begin{bmatrix}k-1&k\\ k-2&k+1\end{bmatrix}, then what is det(A_1)+det(A_2)+det(A_3)+\dots+det(A_{100}) equal to?

Explanation

det(A_k) = (k-1)(k+1) - k(k-2) = (k^2 - 1) - (k^2 - 2k) = 2k - 1. The sum is \sum_{k=1}^{100} (2k - 1), which is the sum of the first 100 odd numbers. The sum of the first n odd numbers is n^2, so 100^2 = 10000.

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