The given matrix is:

\[
\begin{bmatrix}
1 & 1 & -3 \\
-1 & 8 & 0 \\
7 & 0 & 0
\end{bmatrix}
\]

To find the **rank** of the matrix, we need to determine the number of **linearly independent rows** or, equivalently, the number of **non-zero rows** in the row echelon form (REF) or reduced row echelon form (RREF).

Let's perform row reduction to convert the matrix into echelon form:

### Step 1: Matrix
\[
\begin{bmatrix}
1 & 1 & -3 \\
-1 & 8 & 0 \\
7 & 0 & 0
\end{bmatrix}
\]

### Step 2: Eliminate the entries below the first pivot (which is 1).
- Add row 1 to row 2.
- Subtract 7 times row 1 from row 3.

\[
\begin{bmatrix}
1 & 1 & -3 \\
0 & 9 & -3 \\
0 & -7 & 21
\end{bmatrix}
\]

### Step 3: Eliminate the entry below the second pivot (which is 9).
- Add \( \frac{7}{9} \times \text{row 2} \) to row 3.

\[
\begin{bmatrix}
1 & 1 & -3 \\
0 & 9 & -3 \\
0 & 0 & 19
\end{bmatrix}
\]

### Step 4: Simplify pivots by dividing rows 2 and 3 by their leading entries.

\[
\begin{bmatrix}
1 & 1 & -3 \\
0 & 1 & -\frac{1}{3} \\
0 & 0 & 1
\end{bmatrix}
\]

### Conclusion:
The matrix is now in echelon form, and we observe three non-zero rows. Therefore, the matrix has **3 pivots** (1 in each row).

### Final Answer:
The **rank** of the matrix is **3**.

---

Would you like further details on any of the steps, or have any questions?

Here are some related questions to expand on the topic:
1. What does it mean for a matrix to have a certain rank?
2. How do you determine if rows are linearly independent?
3. How does the rank relate to the solutions of a system of equations?
4. What is the significance of row echelon form in linear algebra?
5. How does rank affect the invertibility of a matrix?

**Tip:** The rank of a matrix is equal to the number of linearly independent rows or columns, whichever is fewer.

what is the rank of the matrix 1 1 −3, −1 8 0, 7 0 0

Discover how to find the rank of a 3x3 matrix using row reduction and echelon form. This problem involves key linear algebra concepts like matrix rank, row echelon form, and linear independence. Suitable for advanced high school or college-level students.

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