If A_k=[ array cc k-1 & k k-2 & k+1 array ] , then what is (A_1)+...

Since, A_K=\left[$\begin{array}{cc}K-1 & K \\ K-2 & K+1\end{array}$\right]$\begin{aligned} \Rightarrow\left|A_K\right| & =(\mathrm{K}-1)(\mathrm{K}+1)-\mathrm{K}(\mathrm{~K}-2) \\ & =\mathrm{K}^2-1-\mathrm{K}^2+2 \mathrm{~K}=2 \mathrm{~K}-1 \end{aligned}$Now, $\left|\mathrm{A}_1\right|+\left|\mathrm{A}_2\right|+\left|\mathrm{A}_3\right|+\ldots . .+\left|\mathrm{A}_{100}\right|\begin{aligned} & =(2 \times 1-1)+(2 \times 2-1)+(2 \times 3-1)+\ldots . \\ & =1+(2 \times 100-1) \\ & =1+3+5+\ldots .+199 \\ & =\frac{100}{2}(1+199)=100 \times 100=10000 \end{aligned}$$$