The rank of the given matrix is $3$. This is found by converting the matrix into its Reduced Row Echelon Form (RREF) and counting the number of non-zero rows.

Would you like further details on the steps involved in achieving the Reduced Row Echelon Form, or do you have any additional questions?

Here are some related questions for deeper understanding:

1. What is the difference between row echelon form and reduced row echelon form?
2. How do you compute the nullity of a matrix using its rank?
3. Can you explain how rank relates to the linear independence of matrix rows or columns?
4. What is the significance of having a full rank in a square matrix?
5. How does the rank of a matrix affect the solutions to a system of linear equations?

Tip: The rank of a matrix gives insight into the dimension of the vector space spanned by its rows or columns.