To find the rank of matrix \( A \), we analyze its row-reduced form, which we obtained earlier. Let's summarize the key points:

### The Matrix \( A \):
\[
A =
\begin{bmatrix}
1 & 0 & -3 \\
1 & 1 & -1 \\
0 & -1 & a - 4 \\
0 & 1 & 2
\end{bmatrix}
\]

### Step 1: Row Reduction (Recap)
The row-reduced form of \( A \), depending on the value of \( a \), is:
\[
A_{rref} =
\begin{bmatrix}
1 & 0 & -3 \\
0 & 1 & 2 \\
0 & 0 & a-2 \\
0 & 0 & 4
\end{bmatrix}
\]

### Step 2: Determine the Rank of \( A \)

The rank of \( A \) is determined by the number of non-zero rows (or the number of pivot positions) in its row-reduced form.

#### Case 1: \( a \neq 2 \)
- When \( a \neq 2 \), the third row \( [0 \ 0 \ a-2] \) has a non-zero entry in the third column.
- Thus, all three rows (excluding the fourth row) contain pivots, meaning the rank of \( A \) is **3**.

#### Case 2: \( a = 2 \)
- When \( a = 2 \), the third row becomes \( [0 \ 0 \ 0] \), making it a zero row.
- This leaves only the first two rows with pivots, meaning the rank of \( A \) is **2**.

### Final Answer:

- **If \( a \neq 2 \):** The rank of \( A \) is **3**.
- **If \( a = 2 \):** The rank of \( A \) is **2**.

Would you like more details or have any questions? Here are some related questions to consider:

1. How does the value of \( a \) specifically affect the rank of matrix \( A \)?
2. What is the relationship between the rank of a matrix and its nullity?
3. How does the rank of a matrix determine the dimension of its column space?
4. Why does a zero row in the row-reduced form indicate a reduction in rank?
5. How would you find the null space of \( A \) for different values of \( a \)?

**Tip:** The rank of a matrix is a fundamental concept that directly relates to the solutions of linear systems and the dimensions of various vector spaces associated with the matrix.

Learn how to determine the rank of matrix A, which depends on the variable a. This problem covers matrix theory, row reduction, and concepts from linear algebra. Suitable for undergraduate students studying linear algebra or matrix theory.

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