Mathematics - Definite Integration Question with Solution | TestHub

$\because A=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ 0& a_{22}& a_{23}\\ 0& 0& a_{33}\end{array}\right]\Rightarrow \left|A\right|=a_{11}{a}_{22}{a}_{33}$

$\Rightarrow \left|\mathrm{A}\right|=\left({\mathrm{J}}_{7,3}-{\mathrm{J}}_{4,3}\right)·\left({\mathrm{J}}_{8,3}-{\mathrm{J}}_{5,3}\right)·\left({\mathrm{J}}_{9,3}-{\mathrm{J}}_{6,3}\right)$

$={\int }_0^{1/2}\left(\frac{x^7-x^4}{x^3-1}\right)dx·{\int }_0^{1/2}\left(\frac{x^8-x^5}{x^3-1}\right)dx·{\int }_0^{1/2}\left(\frac{x^9-x^6}{x^3-1}\right)dx$

$={\int }_0^{1/2}{x}^{4}dx·{\int }_0^{1/2}{x}^{5}dx·{\int }_0^{1/2}{x}^{6}dx$

$=\frac{1}{\left(210\right)·(2{)}^{18}}$

Now

$\left|adj{A}^{-1}\right|={\left|A^{-1}\right|}^2=\frac{1}{|A{|}^2}={\left(\left(210\right)·\left(2^{18}\right)\right)}^2=(105{)}^2·(2{)}^{38}$