The sum of three numbers is $6$. Thrice the third number when added to the first number gives $7$. On adding three times the first number to the sum of the second and third numbers,we get $12$. The product of these numbers is:

If $x=a, y=b, z=c$ is the solution of the system of simultaneous linear equations $x+y+z=4$,$x-y+z=2$,and $x+2y+2z=1$,then $ab+bc+ca=$

Let $A = \begin{bmatrix} i & -i \\ -i & i \end{bmatrix}$,where $i = \sqrt{-1}$. Then,the system of linear equations $A^{8} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 8 \\ 64 \end{bmatrix}$ has :

If the matrix equation is given,then $x=$ . . . . . . and $y=$ . . . . . . .

Consider the system of equations:
$x - 2y + 3z = -1$; $-x + y - 2z = k$; $x - 3y + 4z = 1$
$STATEMENT-1$: The system of equations has no solution for $k \neq 3$.
$STATEMENT-2$: The determinant $\left|\begin{array}{ccc}1 & -2 & 3 \\ -1 & 1 & -2 \\ 1 & -3 & 4\end{array}\right| = 0$.

If the following system of linear equations
$2x + y + z = 5$
$x - y + z = 3$
$x + y + az = b$
has no solution,then :