Rank of matrix:

6 1 3 8
4 2 6 -1
10 3 9 7
16 4 12 15

To find the rank of a matrix, we need to reduce the matrix to its row echelon form (or reduced row echelon form) and count the number of non-zero rows. Let's proceed step by step.

The given matrix is:

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

We'll use Gaussian elimination to reduce it to row echelon form.

### Step 1: Eliminate the elements below the first pivot (6 in the first row).

1. Perform \( R_2 \leftarrow R_2 - \frac{4}{6}R_1 \) on the second row.
2. Perform \( R_3 \leftarrow R_3 - \frac{10}{6}R_1 \) on the third row.
3. Perform \( R_4 \leftarrow R_4 - \frac{16}{6}R_1 \) on the fourth row.

The new matrix becomes:

\[
\begin{bmatrix}
6 & 1 & 3 & 8 \\
0 & 1.33 & 3.67 & -6.33 \\
0 & 1.33 & 4 & -6.33 \\
0 & 1.33 & 4 & -6.33
\end{bmatrix}
\]

### Step 2: Eliminate the elements below the second pivot (1.33 in the second row).

Since the third and fourth rows are identical to the second row, we can directly proceed to the final step.

### Step 3: Count the non-zero rows.

The row echelon form will have the following non-zero rows:

\[
\begin{bmatrix}
6 & 1 & 3 & 8 \\
0 & 1.33 & 3.67 & -6.33
\end{bmatrix}
\]

There are 2 non-zero rows.

Thus, **the rank of the matrix is 2**.

Would you like more details or have any questions about this process?

### Related Questions:
1. How do we determine if a matrix is singular using its rank?
2. What is the significance of row echelon form in solving systems of equations?
3. How does the number of non-zero rows relate to the dimension of the row space?
4. What are the applications of the rank of a matrix in real-world problems?
5. How can we find the nullity of this matrix using the rank-nullity theorem?

### Tip:
Always check the determinant of a square matrix; if it's non-zero, the matrix is full-rank!

Learn how to find the rank of a 4x4 matrix using Gaussian elimination. This detailed step-by-step solution covers matrix reduction to row echelon form and calculating the number of non-zero rows to determine rank. Suitable for undergraduate students studying linear algebra.

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