For what value of $\lambda$,the system of equations $x + y + z = 6$,$x + 2y + 3z = 10$,and $x + 2y + \lambda z = 12$ is inconsistent? $\lambda = $ ........

Solve the system of linear equations using the matrix method: $5x + 2y = 4$ and $7x + 3y = 5$.

The sum of three numbers is $6$. If we multiply the third number by $3$ and add the second number to it,we get $11$. By adding the first and third numbers,we get double the second number. Represent this algebraically and find the numbers using the matrix method.

For $\alpha, \beta \in [0, 2\pi]$ and $\gamma \in [0, \pi)$, consider the system of equations:
$2 \sin \alpha - \cos \beta + 3 \tan \gamma = 3$
$4 \sin \alpha + 2 \cos \beta - 2 \tan \gamma = 2$
$6 \sin \alpha - 3 \cos \beta + \tan \gamma = 9$
Then, which one of the following is true?

For the equations $x+2y+3z=1$,$2x+y+3z=2$,and $5x+5y+9z=4$,which of the following is true?

For a matrix $A=\begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{bmatrix}$, if $U_{1}, U_{2}$, and $U_{3}$ are $3 \times 1$ column matrices satisfying $A U_{1}=\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$, $A U_{2}=\begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix}$, $A U_{3}=\begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix}$, and $U$ is a $3 \times 3$ matrix whose columns are $U_{1}, U_{2}$, and $U_{3}$, then the sum of the elements of $U^{-1}$ is: