find the rank of the matrix with the normal method
[1 2 1 0
-2 4 3 0
1 0 2 -8]

To find the rank of a matrix using the normal method, we need to perform row operations to reduce the matrix to its row-echelon form or its reduced row-echelon form. The rank of the matrix is the number of non-zero rows in this form. Let's find the rank of the given matrix step-by-step.

The matrix is:

\[
A = \begin{bmatrix}
1 & 2 & 1 & 0 \\
-2 & 4 & 3 & 0 \\
1 & 0 & 2 & -8
\end{bmatrix}
\]

### Step 1: Perform Row Operations to Achieve Row-Echelon Form

1. **Make the first element of the first row a pivot**: It is already a pivot (1).

2. **Eliminate the elements below the pivot in the first column**:
   - For Row 2: Add 2 times Row 1 to Row 2.
   - For Row 3: Subtract Row 1 from Row 3.

\[
\begin{bmatrix}
1 & 2 & 1 & 0 \\
0 & 8 & 5 & 0 \\
0 & -2 & 1 & -8
\end{bmatrix}
\]

3. **Make the second element of the second row a pivot**: Divide Row 2 by 8.

\[
\begin{bmatrix}
1 & 2 & 1 & 0 \\
0 & 1 & \frac{5}{8} & 0 \\
0 & -2 & 1 & -8
\end{bmatrix}
\]

4. **Eliminate the element below and above the pivot in the second column**:
   - For Row 1: Subtract 2 times Row 2 from Row 1.
   - For Row 3: Add 2 times Row 2 to Row 3.

\[
\begin{bmatrix}
1 & 0 & \frac{-1}{4} & 0 \\
0 & 1 & \frac{5}{8} & 0 \\
0 & 0 & \frac{21}{4} & -8
\end{bmatrix}
\]

5. **Make the third element of the third row a pivot**: Divide Row 3 by \(\frac{21}{4}\).

\[
\begin{bmatrix}
1 & 0 & \frac{-1}{4} & 0 \\
0 & 1 & \frac{5}{8} & 0 \\
0 & 0 & 1 & \frac{-32}{21}
\end{bmatrix}
\]

6. **Eliminate the elements above the pivot in the third column**:
   - For Row 1: Add \(\frac{1}{4}\) times Row 3 to Row 1.
   - For Row 2: Subtract \(\frac{5}{8}\) times Row 3 from Row 2.

\[
\begin{bmatrix}
1 & 0 & 0 & \frac{-8}{21} \\
0 & 1 & 0 & \frac{80}{21} \\
0 & 0 & 1 & \frac{-32}{21}
\end{bmatrix}
\]

### Step 2: Determine the Rank

The matrix is now in row-echelon form with three non-zero rows. Therefore, the rank of the matrix is:

\[
\boxed{3}
\]

Would you like a detailed explanation or have any questions?

### Relative Questions:
1. How can the rank of a matrix be interpreted geometrically?
2. What are the different methods for determining the rank of a matrix?
3. How does the row-echelon form differ from the reduced row-echelon form?
4. What is the relationship between the rank of a matrix and its determinant?
5. How does the rank of a matrix relate to the solutions of a system of linear equations?

**Tip:** When performing row operations, always aim to create zeros below and above each pivot to simplify the process of finding the rank.

find the rank of the matrix with the normal method [1 2 1 0 -2 4 3 0 1 0 2 -8]

Discover how to find the rank of a matrix using the normal method through step-by-step row operations. Learn about row echelon form, Gaussian elimination, and how the rank is determined by the number of non-zero rows. Suitable for Grades 11-12 and college-level students.

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