Question

If $A_i=\left[\begin{array}{cc}{a}^i& b^i\\ b^i& a^i\end{array}\right]$ and $\left|a\right|<1,\left|b\right|<1$ then ${\sum }_{i=1}^{\infty }\left|A_i\right|$ is equal to

Answer 1

$|A{|}_i=\left|\begin{array}{cc}{a}^i& b^i\\ b^i& a^i\end{array}\right|=a^{2i}-b^{2i}$

${\sum }_{i=1}^{\infty }|A_i|=(a^2-b^2)+(a^4-b^4)+(a^6-b^6)+....$

$=(a^2+a^4+a^6+....\infty )-(b^2+b^4+b^6+...\infty )$

$=\frac{a^2}{1-a^2}-\frac{b^2}{1-b^2}$

$=\frac{a^2-b^2}{(1-a^2)(1-b^2)}$