If $A = \begin{bmatrix} 2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2 \end{bmatrix}$,find $A^{-1}$. Using $A^{-1}$,solve the system of equations: $2x - 3y + 5z = 11$,$3x + 2y - 4z = -5$,and $x + y - 2z = -3$.

Find the matrix $X$ such that $X \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} = \begin{bmatrix} -7 & -8 & -9 \\ 2 & 4 & 6 \end{bmatrix}$.

Let $A=\left[\begin{array}{lll}2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1\end{array}\right]$,$B=\left[B_1, B_2, B_3\right]$,where $B_1, B_2, B_3$ are column matrices,and $AB_1=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$,$AB_2=\left[\begin{array}{l}2 \\ 3 \\ 0\end{array}\right]$,$AB_3=\left[\begin{array}{l}3 \\ 2 \\ 1\end{array}\right]$. If $\alpha=|B|$ and $\beta$ is the sum of all the diagonal elements of $B$,then $\alpha^3+\beta^3$ is equal to

If the system of simultaneous linear equations $x-2y+z=0$, $2x+3y+z=6$, and $x+2y+pz=q$ has infinitely many solutions, then:

The cost of $4 \, kg$ onion,$3 \, kg$ wheat and $2 \, kg$ rice is $Rs \, 60$. The cost of $2 \, kg$ onion,$4 \, kg$ wheat and $6 \, kg$ rice is $Rs \, 90$. The cost of $6 \, kg$ onion,$2 \, kg$ wheat and $3 \, kg$ rice is $Rs \, 70$. Find the cost of each item per $kg$ using the matrix method.

The positive value of $a$ for which the system of linear homogeneous equations $x+ay+z=0$, $ax+2y-z=0$, and $2x+3y+z=0$ has non-trivial solutions is