The system of equations $a + b - 2c = 0$,$2a - 3b + c = 0$,and $a - 5b + 4c = \alpha$ is consistent for $\alpha$ equal to

Examine the consistency of the system of equations: $3x - y - 2z = 2$; $2y - z = -1$; $3x - 5y = 3$.

The solution of equations $x + y = 10$,$2x + y = 18$,and $4x - 3y = 26$ is:

The number of $3 \times 3$ matrices $A$ whose entries are either $0$ or $1$ and for which the system $A\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$ has exactly two distinct solutions,is

If $x = a$,$y = b$,$z = c$ is a solution of the system of linear equations $x + 8y + 7z = 0$,$9x + 2y + 3z = 0$,and $x + y + z = 0$ such that the point $(a, b, c)$ lies on the plane $x + 2y + z = 6$,then $2a + b + c$ equals

The system of equations $\lambda x - y + (\cos\theta) z = 0$,$3x + y + 2z = 0$,and $(\cos\theta) x + y + 2z = 0$ for $0 < \theta < 2\pi$ has non-trivial solution$(s)$: