Find rank of 3*3 matrix  7 -1 0    1  1  4          13 -3 -4

Let's denote the matrix as \( A \):

\[
A = \begin{pmatrix} 
7 & -1 & 0 \\ 
1 & 1 & 4 \\ 
13 & -3 & -4 
\end{pmatrix}
\]

### Step 1: Convert the matrix to row echelon form (REF)

We perform elementary row operations to bring the matrix into upper triangular (REF) form.

Start with matrix \( A \):

\[
A = \begin{pmatrix} 
7 & -1 & 0 \\ 
1 & 1 & 4 \\ 
13 & -3 & -4 
\end{pmatrix}
\]

- First, make the pivot in the first column equal to 1 by performing \( R_2 \leftarrow R_2 - \frac{1}{7}R_1 \) and \( R_3 \leftarrow R_3 - \frac{13}{7}R_1 \).

This yields:

\[
A = \begin{pmatrix} 
7 & -1 & 0 \\ 
0 & \frac{8}{7} & 4 \\ 
0 & \frac{-4}{7} & -4 
\end{pmatrix}
\]

### Step 2: Continue with row reduction

- Now, make the pivot in the second column a 1 by dividing \( R_2 \) by \( \frac{8}{7} \).

\[
A = \begin{pmatrix} 
7 & -1 & 0 \\ 
0 & 1 & \frac{7}{2} \\ 
0 & \frac{-4}{7} & -4 
\end{pmatrix}
\]

- Finally, eliminate the value in the second row, third column by performing \( R_3 \leftarrow R_3 + \frac{4}{7}R_2 \).

\[
A = \begin{pmatrix} 
7 & -1 & 0 \\ 
0 & 1 & \frac{7}{2} \\ 
0 & 0 & 0 
\end{pmatrix}
\]

### Step 3: Determine the rank

The matrix is now in row echelon form. Since there are 2 non-zero rows, the rank of the matrix is:

\[
\text{Rank}(A) = 2
\]

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Would you like more details on how row echelon form works or have any other questions?

Here are some related questions to explore:
1. What are elementary row operations in matrices?
2. How do you determine if a matrix is in row echelon form (REF)?
3. How does the rank of a matrix relate to its solutions in linear equations?
4. What is the difference between row echelon form and reduced row echelon form?
5. Can you explain how the rank affects the invertibility of a matrix?

**Tip:** The rank of a matrix represents the maximum number of linearly independent rows or columns.

Discover how to determine the rank of a 3x3 matrix with a step-by-step solution. This problem covers matrix operations and row echelon form, essential in linear algebra. Suitable for advanced high school or college students.

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