If the system of equations $kx + 2y - z = 2, (k - 1)x + ky + z = 1, x + (k - 1)y + kz = 3$ has only one solution,then the number of possible real value$(s)$ of $k$ is -

In a legislative assembly election,a political group hired a public relations firm to promote its candidate in three ways: telephone,house calls,and letters. The cost per contact (in paise) is given in matrix $A$ as $A = \begin{bmatrix} 40 \\ 100 \\ 50 \end{bmatrix} \begin{matrix} \text{Telephone} \\ \text{Housecall} \\ \text{Letter} \end{matrix}$. The number of contacts of each type made in two cities $X$ and $Y$ is given by $B = \begin{bmatrix} 1000 & 500 & 5000 \\ 3000 & 1000 & 10000 \end{bmatrix} \begin{matrix} \text{Telephone} & \text{Housecall} & \text{Letter} \\ \to X \\ \to Y \end{matrix}$. Find the total amount spent by the group in the two cities $X$ and $Y$.

Statement $1$: If the system of equations $x + ky + 3z = 0, 3x + ky - 2z = 0, 2x + 3y - 4z = 0$ has a nontrivial solution,then the value of $k$ is $\frac{31}{2}$.
Statement $2$: $A$ system of three homogeneous equations in three variables has a nontrivial solution if the determinant of the coefficient matrix is zero.

The system of equations $(\sin\theta) x + 2z = 0$,$(\cos\theta) x + (\sin\theta) y = 0$,and $(\cos\theta) y + 2z = a$ has

If the system of equations $x+y+z=1$,$x+2y+4z=k$ and $x+4y+10z=k^2$ is consistent,then $k$ is equal to

Given $A = \begin{bmatrix} 1 & 3 \\ 2 & 2 \end{bmatrix}$ and $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. If $A - \lambda I$ is a singular matrix,then: