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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.

The reduced row echelon form of a system of linear equations is given by the matrix:
\[
\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix}
\]
Determine whether the system has a unique solution, infinitely many solutions, or no solutions.

Analyze the reduced row echelon form (RREF) matrix to determine whether a system of linear equations has a unique solution, infinitely many solutions, or no solutions. In this case, the system leads to infinitely many solutions with a free variable.

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