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:

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.