Let $A$ and $B$ be any two $n \times n$ matrices such that the following conditions hold: $A B=B A$ and there exist positive integers $k$ and $l$ such that $A^k=I$ ( the identity matrix) and $B^l=0$ (the zero matrix). Then,

If the matrix $AB = O$, then

Construct a $2 \times 2$ matrix, $A=\left[a_{ij}\right]$, whose elements are given by : $a_{i j}=\frac{(i+j)^{2}}{2}$

Let $p$ , $q$ , $r$ are three real numbers satisfying $\left[ {p\,\,q\,\,r} \right]\left[ {\begin{array}{*{20}{c}}
  2&p&q \\ 
  { – 3}&q&{ – p + r} \\ 
  {12}&r&{ – q + 3r} 
\end{array}} \right] = \left[ {5\,\,\,b\,\,c} \right]$ , then minimum value of $(b + c)$ is 

Let $A=\left[a_{i j}\right]$ be a square matrix of order $3$ such that $a_{i j}=2^{j-i}$, for all $i, j=1,2,3$. Then, the matrix $A ^{2}+ A ^{3}+\ldots+ A ^{10}$ is equal to