If $A = \begin{bmatrix} 1 & 0 & 2 \\ 2 & 1 & 3 \\ 3 & 2 & 4 \end{bmatrix}$,then $A^2 - 5A + 6I =$

Let $S = \left\{ \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} : a_{ij} \in \{0, 1, 2\}, a_{11} = a_{22} \right\}$. Then the number of non-singular matrices in the set $S$ is

Which of the following matrices can $NOT$ be obtained from the matrix $\left[\begin{array}{cc}-1 & 2 \\ 1 & -1\end{array}\right]$ by a single elementary row operation?

Two farmers,Ramkishan and Gurcharan Singh,cultivate only three varieties of rice: Basmati,Permal,and Naura. The sales (in Rupees) of these varieties of rice by both farmers in the months of September and October are given by the following matrices $A$ and $B$.
September Sales (in Rupees)
$A = \begin{bmatrix} \text{Basmati} & \text{Permal} & \text{Naura} \\ 10,000 & 20,000 & 30,000 \\ 50,000 & 30,000 & 10,000 \end{bmatrix} \begin{matrix} \\ \text{Ramkishan} \\ \text{Gurcharan Singh} \end{matrix}$
October Sales (in Rupees)
$B = \begin{bmatrix} \text{Basmati} & \text{Permal} & \text{Naura} \\ 5,000 & 10,000 & 6,000 \\ 20,000 & 10,000 & 10,000 \end{bmatrix} \begin{matrix} \\ \text{Ramkishan} \\ \text{Gurcharan Singh} \end{matrix}$
$(i)$ 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.

If the matrix $\left[\begin{array}{rr}2 & 3 \\ 5 & -1\end{array}\right]=A+B$,where $A$ is symmetric and $B$ is skew-symmetric,then $B$ is equal to

Find $X$ and $Y,$ if $2X+3Y=\left[\begin{array}{cc}2 & 3 \\ 4 & 0\end{array}\right]$ and $3X+2Y=\left[\begin{array}{cc}2 & -2 \\ -1 & 5\end{array}\right].$