question_answer

If three matrices \[A=\left[ \begin{matrix}    2 & 1  \\    4 & 1  \\ \end{matrix} \right],B=\left[ \begin{matrix}    3 & 4  \\    2 & 3  \\ \end{matrix} \right]\] and\[C=\left[ \begin{matrix}    3 & -4  \\    -2 & 3  \\ \end{matrix} \right]\]. Then \[{{t}_{r}}(A)+{{t}_{r}}\left( \frac{ABC}{2} \right)+{{t}_{r}}\left( \frac{A{{(BC)}^{2}}}{4} \right)+{{t}_{r}}\left( \frac{A{{(BC)}^{3}}}{8} \right)\]\[+.....\infty =\,\,\,\,\,\,+\]

Answer:

6.00