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To solve this problem, we need to determine which Gaussian elimination step is appropriate for the first column of the given matrix:

\[
\begin{bmatrix}
1 & 2 & 3 \\
4 & 1 & 6 \\
5 & 0 & 1
\end{bmatrix}
\]

Gaussian elimination works by transforming the matrix into row-echelon form, which typically involves getting a leading 1 in each pivot position and zeros below it in the same column.

### Step-by-Step Approach:
We start by working on the first column (the "pivot" column). The first entry in this column (row 1, column 1) is already 1, so no changes are needed for that row.

Next, we need to eliminate the entries below this pivot (the 4 and 5 in rows 2 and 3, respectively).

#### Options:
1. **Option 1**: Perform \( \frac{1}{4} R_2 \to R_2 \). This step is about scaling the second row by \( \frac{1}{4} \), but that doesn't directly help us eliminate the 4 in the second row, first column.

2. **Option 2**: Perform \( -2 R_2 + R_1 \to R_1 \). This option seems to be trying to zero out an element in the first column, but it mixes rows 1 and 2 incorrectly and wouldn't help with eliminating the 4.

3. **Option 3**: Perform \( -4 R_1 + R_2 \to R_2 \). This operation subtracts 4 times the first row from the second row, which would turn the 4 in the second row, first column into 0 (because \( 4 - 4 \times 1 = 0 \)). This is the correct first step for Gaussian elimination in the first column.

4. **Option 4**: Perform \( \frac{1}{5} R_3 \to R_3 \). This step involves scaling the third row by \( \frac{1}{5} \), but this doesn't eliminate the 5 in the first column.

#### Conclusion:
The correct first step is **Option 3**: Perform \( -4 R_1 + R_2 \to R_2 \) to zero out the 4 in the second row, first column.

Let me know if you need more details or clarifications!

### Follow-up Questions:
1. What happens after eliminating the 4 in the second row, first column?
2. How would you proceed to eliminate the 5 in the third row, first column?
3. How can we perform Gaussian elimination to fully convert the matrix to row-echelon form?
4. What are the general rules for Gaussian elimination in terms of pivot and row operations?
5. How does Gaussian elimination help in solving systems of linear equations?

### Tip:
Always start by focusing on one pivot column and eliminate entries below the pivot before moving to the next column

To perform Gaussian elimination, which of the following must be done first for the first column of the matrix [ [1, 2, 3], [4, 1, 6], [5, 0, 1] ]?

This problem covers the process of Gaussian elimination for a 3x3 matrix, focusing on the first step for transforming the matrix into row-echelon form. Learn about key linear algebra concepts like elementary row operations and the Gaussian elimination method.

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