Question

$A_{3\times 3}$ is a matrix such that $|A|=a,B=\left(adjA\right)$ such that $|B|=b$. Find the value of $\frac{(a{b}^2+a^{2}b+1)S}{25}$ where $\frac{1}{2}S=\frac{a}{b}+\frac{a^2}{b^3}+\frac{a^3}{b^5}+.........upto\infty$, and $a=3$.

Answer 1

$|B|=|adjA|=|A{|}^2=9$
$\frac{S}{2}=\frac{a/b}{1-a/b^2}=\frac{ab}{b^2-a}=\frac{27}{81-3}=\frac{27}{78}=\frac{9}{26}$
$\therefore (a{b}^2+a^{2}b+1)S=225$