If $(x, y, z)=(\alpha, \beta, \gamma)$ is the unique solution of the system of simultaneous linear equations $3x - 4y + z + 7 = 0$, $2x + 3y - z = 10$, and $x - 2y - 3z = 3$, then $\alpha = $

Under which of the following condition$(s)$ does the system of equations $\begin{bmatrix} 1 & 2 & 4 \\ 2 & 1 & 2 \\ 1 & 2 & a-4 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 6 \\ 4 \\ a \end{bmatrix}$ possess a unique solution?

Examine the consistency of the system of equations: $x+3y=5$ and $2x+6y=8$.

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.

For a real number $\alpha$,if the system of linear equations $\begin{bmatrix} 1 & \alpha & \alpha^2 \\ \alpha & 1 & \alpha \\ \alpha^2 & \alpha & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix}$ has infinitely many solutions,then $1+\alpha+\alpha^2=$

For the system $S$ of linear equations $x+y+z=3, 2x+2y-z=3, x+y+\lambda z=1$, the incorrect option among the following statements is: