For how many different values of $a$ does the following system of linear equations have at least two distinct solutions?
$ax + y = 0$
$x + (a + 10)y = 0$

If the system of linear equations: $x + y + z = 6, x + 2y + 5z = 10, 2x + 3y + \lambda z = \mu$ has infinitely many solutions,then the value of $\lambda + \mu$ equals:

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:

The system of equations $x+2y+3z=6$,$x+3y+5z=9$,and $2x+5y+az=12$ has no solution when $a=$

If the system of equations $x+4y-z=\lambda$,$7x+9y+\mu z=-3$,and $5x+y+2z=-1$ has infinitely many solutions,then $(2\mu+3\lambda)$ is equal to:

If the system of simultaneous linear equations $x+\lambda y-2 z=1$,$x-y+\lambda z=2$,and $x-2 y+3 z=3$ is inconsistent for $\lambda=\lambda_1$ and $\lambda_2$,then $\lambda_1+\lambda_2=$