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To find the rank of the matrix using the Echelon form method, we must row-reduce the matrix to its echelon form. The matrix given is:

\[
\begin{bmatrix}
2 & -4 & 3 & -1 & 0 \\
1 & -2 & -1 & -4 & 2 \\
0 & 1 & 3 & 5 & 1 \\
4 & -7 & 4 & -4 & 5
\end{bmatrix}
\]

Let's perform the row-reducing steps:

### Step 1: Make the first pivot element (row 1, column 1) equal to 1.
To do this, we divide the first row by 2.

\[
\text{R1} \rightarrow \frac{1}{2} \times \text{R1} = \begin{bmatrix} 1 & -2 & 1.5 & -0.5 & 0 \end{bmatrix}
\]

New matrix:
\[
\begin{bmatrix}
1 & -2 & 1.5 & -0.5 & 0 \\
1 & -2 & -1 & -4 & 2 \\
0 & 1 & 3 & 5 & 1 \\
4 & -7 & 4 & -4 & 5
\end{bmatrix}
\]

### Step 2: Eliminate the elements below the pivot (make the elements in column 1 below the first row 0).
- Subtract row 1 from row 2: \(\text{R2} \rightarrow \text{R2} - \text{R1}\)
- Subtract 4 times row 1 from row 4: \(\text{R4} \rightarrow \text{R4} - 4\text{R1}\)

New matrix:
\[
\begin{bmatrix}
1 & -2 & 1.5 & -0.5 & 0 \\
0 & 0 & -2.5 & -3.5 & 2 \\
0 & 1 & 3 & 5 & 1 \\
0 & 1 & -2 & -2 & 5
\end{bmatrix}
\]

### Step 3: Make the second pivot (row 3, column 2) equal to 1.
Row 3 already has a pivot of 1 in column 2.

### Step 4: Eliminate the elements below and above the second pivot.
- Subtract row 3 from row 4: \(\text{R4} \rightarrow \text{R4} - \text{R3}\)

New matrix:
\[
\begin{bmatrix}
1 & -2 & 1.5 & -0.5 & 0 \\
0 & 0 & -2.5 & -3.5 & 2 \\
0 & 1 & 3 & 5 & 1 \\
0 & 0 & -5 & -7 & 4
\end{bmatrix}
\]

### Step 5: Make the third pivot (row 2, column 3) equal to 1.
Divide row 2 by \(-2.5\).

New matrix:
\[
\begin{bmatrix}
1 & -2 & 1.5 & -0.5 & 0 \\
0 & 0 & 1 & 1.4 & -0.8 \\
0 & 1 & 3 & 5 & 1 \\
0 & 0 & -5 & -7 & 4
\end{bmatrix}
\]

### Step 6: Eliminate the elements below and above the third pivot.
- Subtract 3 times row 2 from row 3.
- Add 5 times row 2 to row 4.

New matrix:
\[
\begin{bmatrix}
1 & -2 & 0 & -2.6 & 1.2 \\
0 & 0 & 1 & 1.4 & -0.8 \\
0 & 1 & 0 & 0.8 & 3.4 \\
0 & 0 & 0 & 0 & 0
\end{bmatrix}
\]

### Step 7: The matrix is now in row-echelon form.

The number of non-zero rows is 3, which means the **rank of the matrix is 3**.

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Would you like a more detailed explanation of any step?

Here are 5 related questions for deeper understanding:
1. How is row echelon form different from reduced row echelon form (RREF)?
2. What is the significance of the rank of a matrix in linear algebra?
3. How can the rank of a matrix help in solving systems of linear equations?
4. Can the rank of a matrix change if its rows or columns are interchanged?
5. How do elementary row operations affect the rank of a matrix?

**Tip:** The rank of a matrix can also be interpreted as the dimension of the row space or column space of the matrix.

Apply Echelon form method to find the rank of the matrix: [[2, -4, 3, -1, 0], [1, -2, -1, -4, 2], [0, 1, 3, 5, 1], [4, -7, 4, -4, 5]]

Learn how to find the rank of the matrix [[2, -4, 3, -1, 0], [1, -2, -1, -4, 2], [0, 1, 3, 5, 1], [4, -7, 4, -4, 5]] using the Echelon form method. This step-by-step solution includes key linear algebra concepts such as row operations and matrix rank, suitable for students in high school or college-level linear algebra.

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