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If ${A_i} = \left[ {\begin{array}{*{20}{c}}{{a^i}}&{{b^i}}\\{{b^i}}&{{a^i}}\end{array}} \right]$and if $|a|\, < 1,\,|b|\, < 1$, then $\sum\limits_{i = 1}^\infty {\det ({A_i})} $is equal to

• A
  $\frac{{{a^2}}}{{{{(1 - a)}^2}}} - \frac{{{b^2}}}{{{{(1 - b)}^2}}}$
• ✓
  $\frac{{{a^2} - {b^2}}}{{(1 - {a^2})(1 - {b^2})}}$
• C
  $\frac{{{a^2}}}{{{{(1 - a)}^2}}} + \frac{{{b^2}}}{{{{(1 - b)}^2}}}$
• D
  $\frac{{{a^2}}}{{{{(1 + a)}^2}}} - \frac{{{b^2}}}{{{{(1 + b)}^2}}}$