If $A=\begin{bmatrix}1&cot~x\\ -cot~x&1\end{bmatrix}$ show that $A^{\prime}A^{-1}=\begin{bmatrix}-cos~2x&-sin~2x\\ sin~2x&-cos~2x\end{bmatrix}$

If the area of $\Delta$ ABC with vertices $A(3,1)$, $B(-2,1)$ and $C(0,k)$ is 5 sq. units, then values of k are:

A man goes to buy fruits from the market. The shopkeeper informs him that 4 apples, 3 oranges and 2 bananas cost ₹ 60; 2 apples, 4 oranges and 6 bananas cost ₹ 90; whereas 6 apples, 2 oranges and 3 bananas cost ₹ 70. Using matrix method, find the cost of one fruit of each kind.

A school wants to allocate students into three clubs Sports, Music and Drama, under following conditions: The number of students in Sports club should be equal to the sum of the number of students in Music and Drama club. The number of students in Music club should be 20 more than half the number of students in Sports club. The total number of students to be allocated in all three clubs are 180. Find the number of students allocated to different clubs, using matrix method.

1. If $x\begin{bmatrix}1\\ 2\end{bmatrix}+y\begin{bmatrix}2\\ 5\end{bmatrix}=\begin{bmatrix}4\\ 9\end{bmatrix}$, then