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The problem asks to reduce the given matrix to its normal form (Row Echelon Form or Reduced Row Echelon Form) and find its rank. The given matrix is:

\[
\begin{bmatrix}
6 & 1 & 3 & 8 \\
4 & 2 & 6 & -1 \\
10 & 3 & 9 & 7 \\
16 & 4 & 12 & 15
\end{bmatrix}
\]

The operations listed beside the matrix are:
- \( A_4 \rightarrow A_4 - (R_1 + R_3) \)
- \( R_3 \rightarrow R_3 - (R_1 + R_2) \)

Let me perform the row operations and find the rank of the matrix step-by-step.The matrix after performing the row operations is:

\[
\begin{bmatrix}
6 & 1 & 3 & 8 \\
4 & 2 & 6 & -1 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{bmatrix}
\]

Converting this to Row Echelon Form (REF), we have:

\[
\begin{bmatrix}
6 & 1 & 3 & 8 \\
0 & 8 & 24 & -38 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{bmatrix}
\]

The rank of this matrix is **2**, because there are 2 non-zero rows.

Would you like further details or clarifications?

Here are 5 related questions:
1. What are the steps to find the rank of a matrix in general?
2. How do elementary row operations affect the rank of a matrix?
3. Can you explain the difference between row echelon form (REF) and reduced row echelon form (RREF)?
4. How would the rank change if one more non-zero row was present?
5. What is the geometric interpretation of the rank of a matrix?

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

Reduce the following matrix to normal form and find its rank.

Learn how to reduce a matrix to its normal form (row echelon form) and find the matrix rank. This solution covers elementary row operations and explains how to identify the rank of a matrix through step-by-step calculations. Suitable for students in Grades 11-12 and college level.

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