Let ${J_{n,m}} = \int\limits_0^{{1 \over 2}} {{{{x^n}} \over {{x^m} - 1}}dx}$, $\forall$ n > m and n, m $\in$ N. Consider a matrix $A = {[{a_{ij}}]_{3 \times 3}}$ where $${a_{ij}} = \left\{ {\matrix{ {{j_{6 + i,3}} - {j_{i + 3,3}},} & {i \le j} \cr {0,} & {i > j} \cr } } \right.$$. Then $\left| {adj{A^{ - 1}}} \right|$ is :

A = $$\left[ {\matrix{ {a_{11}} &amp; {a_{12}} &amp; {a_{13}} \cr {{a_{21}}} &amp; {{a_{22}}} &amp; {{a_{23}}} \cr {{a_{31}}} &amp; {{a_{32}}} &amp; {{a_{33}}} \cr } } \right]$$<br><br>${J_{6 + i,3}} - {J_{i + 3,3}}; i \le j$<br><br>$$ = \int_0^{{1 \over 2}} {{{{x^{6 + i}}} \over {{x^3} - 1}} - \int_0^{{1 \over 2}} {{{{x^{i + 3}}} \over {{x^3} - 1}}} } $$<br><br>$=\int_0^{1/2} {{{{x^{i + 3}}({x^3} - 1)} \over {{x^3} - 1}}}$<br><br>$$ = {{{x^{3 + i + 1}}} \over {3 + i + 1}} = \left( {{{{x^{4 + i}}} \over {4 + i}}} \right)_0^{1/2}$$<br><br>$\therefore$ $${a_{ij}} = {j_{6 + i,3}} - {j_{i + 3,3}} = {{{{\left( {{1 \over 2}} \right)}^{4 + i}}} \over {4 + i}}$$<br><br>$${a_{11}} = {{{{\left( {{1 \over 2}} \right)}^5}} \over 5} = {1 \over {{{5.2}^5}}}$$<br><br>${a_{12}} = {1 \over {{{5.2}^5}}}$<br><br>${a_{13}} = {1 \over {{{5.2}^5}}}$<br><br>${a_{22}} = {1 \over {{{6.2}^6}}}$<br><br>${a_{23}} = {1 \over {{{6.2}^6}}}$<br><br>${a_{33}} = {1 \over {{{7.2}^7}}}$<br><br>$$A = \left[ {\matrix{ {{1 \over {{{5.2}^5}}}} &amp; {{1 \over {{{5.2}^5}}}} &amp; {{1 \over {{{5.2}^5}}}} \cr 0 &amp; {{1 \over {{{6.2}^6}}}} &amp; {{1 \over {{{6.2}^6}}}} \cr 0 &amp; 0 &amp; {{1 \over {{{7.2}^7}}}} \cr } } \right]$$<br><br>$$\left| A \right| = {1 \over {{{5.2}^5}}}\left[ {{1 \over {{{6.2}^6}}} \times {1 \over {{{7.2}^7}}}} \right]$$<br><br>$\left| A \right| = {1 \over {{{210.2}^{18}}}}$<br><br>$$\left| {adj{A^{ - 1}}} \right| = {\left| {{A^{ - 1}}} \right|^{n - 1}} = {\left| {{A^{ - 1}}} \right|^2} = {1 \over {{{\left( {\left| A \right|} \right)}^2}}}$$<br><br>$= {({210.2^{18}})^2}$<br><br>= ${(105)^2} \times {2^{38}}$