We interpret the chained equality as the system

We need the value(s) of $\lambda$ for which the system has no solution.

1. Form the coefficient matrix

The coefficient matrix is

and the augmented matrix is

For a system to have no solution, we need:

• $\det(A)=0$ so that the system is singular, and
• the augmented system must be inconsistent.

2. Compute $\det(A)$

Expand along the first row:

This would never be zero, which suggests a sign issue in expansion. So let us compute carefully again.

Using direct expansion:

Again impossible to vanish. This means the intended equations are more likely the standard interpretation from the printed question:

Now reduce using the first two equations.

3. Solve first two equations

Add the first two equations:

Substitute into the first equation:

So

4. Use the third equation

Third equation:

With $y=2$ and $z=2x+2$,

Since

for every real $\lambda$, there is always a unique value of $x$. Thus the system can never be inconsistent.

So there must be a typo in the question/options, or the intended third equation likely has coefficient $\lambda$ (not $|\lambda|$) and/or different constants.

5. Check the option matching the usual intended form

A very common version is

Then using $y=2,\ z=2x+2$:

For no solution, we need coefficient of $x$ to be zero but RHS nonzero:

and then RHS $=2(-2)-2=-6\neq 0$, so no solution at $\lambda=-2$. But $-2$ is not among the options.

Hence, as written, none of the options is correct.

6. Conclusion

The given system, with $|\lambda|$ in the third equation, is always consistent and has a unique solution for every real $\lambda$. Therefore it never has no solution.

So the stored answer $\text{B}$ does not agree with the mathematics of the written question.