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

$A=\left[\begin{array}{l}a_{i j}\end{array}\right]_{m\times n}$ is a square matrix, if

Suppose the vectors $x_{1}, x_{2}$ and $x_{3}$ are the solutions of the system of linear equations, $Ax = b$ when the vector $b$ on the right side is equal to $b _{1}, b _{2}$ and $b _{3}$ respectively. If $x =\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right], x _{2}=\left[\begin{array}{l}0 \\ 2 \\ 1\end{array}\right], x _{3}=\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right], b _{1}=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$ $b _{2}=\left[\begin{array}{l}0 \\ 2 \\ 0\end{array}\right]$ and $b _{3}=\left[\begin{array}{l}0 \\ 0 \\ 2\end{array}\right],$ then the determinant of $A$ is equal to

If $A$ is square matrix such that $A^{2}=A$, then $(1+A)^{3}-7 A$ is equal to

Two farmers Ramkishan and Gurcharan Singh cultivates only three varieties of rice namely Basmati, Permal and Naura. The sale (in Rupees) of these varieties of rice by both the farmers in the month of September and October are given by the following matrices $A$ and $B$.

September Sales (in Rupees)

$A=$  $\left[ {\begin{array}{*{20}{c}}
  {{\text{ Basmati }}}&{{\text{ Permal }}}&{{\text{ Naura }}} \\ 
  {10,000}&{20,000}&{30,000} \\ 
  {50,000}&{30,000}&{10,000} 
\end{array}} \right]\,$ $\begin{matrix}
   {}  \\
    \mathrm {Ramkrishan} \,\,\,\,\,\,\,\,\,\,\,\,\,  \\
    \mathrm {Gurcharan}\,\, \mathrm {Singh}  \\
\end{matrix}$

October Sales (in Rupees)

$B=\left[\begin{array}{ccc}\text { Basmati } & \text { Permal } & \text { Naura } \\ 5000 & 10,000 & 6000 \\ 20,000 & 10,000 & 10,000\end{array}\right]$ $\begin{matrix}
   {}  \\
    \mathrm {Ramkrishan} \,\,\,\,\,\,\,\,\,\,\,\,\,  \\
   \mathrm {Gurcharan}\,\,\mathrm {Singh}  \\
\end{matrix}$

$(i)$ Find the combined sales in September and October for each farmer in each variety.

$(ii)$ Find the decrease in sales from September to October.

$(iii)$ If both farmers receive $2\%$ profit on gross sales, compute the profit for each farmer and for each variety sold in October.