Question

Let $I_n=\underset{0}{\overset{1}{\int }}\frac{x^n}{x^{2012}-1}dx,J_n=\underset{0}{\overset{1}{\int }}\frac{x^n}{x^{2013}+1}dx$ for all $n>2012,n\in N$ and matrix $A=(a_{ij}{)}_{3\times 3},$ where $a_{ij}=\{\begin{array}{c}{I}_{2012+i}-I_i,i=j\\ 0,i\ne j\end{array}$
and matrix $B=(b_{ij}{)}_{3\times 3},$
where $b_{ij}=\{\begin{array}{c}{J}_{2016+j}+J_{j+3},i=j\\ 0,i\ne j\end{array}.$
Then the value of $\text{trace}\left(A^{-1}\right)+|B^{-1}|$ is

Answer 1

$I_{2012+i}-I_i={\int }_0^1\frac{x^i(x^{2012}-1)}{(x^{2012}-1)}dx=\frac{1}{1+i}$

$J_{2016+j}+J_{j+3}={\int }_0^1\frac{x^{j+3}(x^{2013}+1)}{(x^{2013}+1)}dx=\frac{1}{j+4}$

$\therefore A=\left[\begin{array}{ccc}\frac{1}{2}& 0& 0\\ 0& \frac{1}{3}& 0\\ 0& 0& \frac{1}{4}\end{array}\right],B=\left[\begin{array}{ccc}\frac{1}{5}& 0& 0\\ 0& \frac{1}{6}& 0\\ 0& 0& \frac{1}{7}\end{array}\right]$

$\text{tr}\left(A^{-1}\right)+|B^{-1}|=(2+3+4)+(5\times 6\times 7)=219$