Let the system of equations $x+2y+3z=5$,$2x+3y+z=9$,and $4x+3y+\lambda z=\mu$ have an infinite number of solutions. Then $\lambda+2\mu$ is equal to:

Let $A = \begin{bmatrix} 1 & 3 \\ 4 & -3 \end{bmatrix}$. Let $S = \left\{ \begin{bmatrix} x \\ y \end{bmatrix} \in \mathbb{R}^2 \mid A \begin{bmatrix} x \\ y \end{bmatrix} = 3 \begin{bmatrix} x \\ y \end{bmatrix} \right\}$. What is the cardinality of $S$?

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 $\lambda_1+\lambda_2=$

If the homogeneous system of linear equations $x-2y+3z=0, 2x+4y-5z=0, 3x+\lambda y+\mu z=0$ has a non-trivial solution,then $8\mu+11\lambda=$

Let $X = \begin{bmatrix} a \\ b \\ c \end{bmatrix}$,$A = \begin{bmatrix} 1 & -1 & 2 \\ 2 & 0 & 1 \\ 3 & 2 & 1 \end{bmatrix}$,and $B = \begin{bmatrix} 3 \\ 1 \\ 4 \end{bmatrix}$. If $AX = B$,then the value of $2a - 3b + 4c$ is:

If $A = \begin{bmatrix} 2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2 \end{bmatrix}$,find $A^{-1}$. Using $A^{-1}$,solve the system of equations: $2x - 3y + 5z = 11$,$3x + 2y - 4z = -5$,and $x + y - 2z = -3$.