Infinitely Many Solutions of Linear Equations System

To find the value of $\lambda + \mu$ for which the system of equations has infinitely many solutions, we can write the given system in augmented matrix form $[A | B]$:

$$\begin{bmatrix} 1 & 1 & 1 & | & 5 \\ 1 & 2 & 3 & | & 9 \\ 1 & 3 & \lambda & | & \mu \end{bmatrix}$$

We apply elementary row operations to transform the matrix into row echelon form:

1. Perform $R_2 \to R_2 - R_1$ and $R_3 \to R_3 - R_1$:

$$\begin{bmatrix} 1 & 1 & 1 & | & 5 \\ 0 & 1 & 2 & | & 4 \\ 0 & 2 & \lambda - 1 & | & \mu - 5 \end{bmatrix}$$

1. Perform $R_3 \to R_3 - 2R_2$:

$$\begin{bmatrix} 1 & 1 & 1 & | & 5 \\ 0 & 1 & 2 & | & 4 \\ 0 & 0 & \lambda - 5 & | & \mu - 13 \end{bmatrix}$$

For the system of linear equations to have infinitely many solutions, the rank of the coefficient matrix $A$ must be equal to the rank of the augmented matrix $[A | B]$, and both must be less than the number of variables (which is 3).

This requires the third row of the augmented matrix to be completely zero:

$$\lambda - 5 = 0 \implies \lambda = 5$$ $$\mu - 13 = 0 \implies \mu = 13$$

Now, calculating the value of $\lambda + \mu$:

$$\lambda + \mu = 5 + 13 = 18$$

Hence, the correct option is B.