The given matrix represents a linear system in its reduced row echelon form (RREF):

$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix}.$

This corresponds to the system of linear equations:

1. $x = 0$,
2. $y = 0$,
3. $0 = 0$ (which is a trivial equation that does not affect the solution set).

From this system, we see that there are specific solutions for $x$ and $y$ (both equal to zero), but there is no equation constraining $z$. Therefore, $z$ is a free variable, which means there are infinitely many solutions.

Based on the options provided, the correct choice is:

B. Infinitely many solutions.

Would you like further details or have any questions?

Here are 5 related questions to further your understanding:

1. What is the general form of the solution for a system when there are infinitely many solutions?
2. How can we identify free variables in a system of linear equations?
3. What does it mean if a row in the RREF is entirely zeros?
4. How does the rank of a matrix relate to the number of solutions of a system of linear equations?
5. What conditions lead to a system having no solutions?

Tip: In a system of linear equations, if you have a free variable, the system generally has infinitely many solutions.