Transform `[(1,2,4),(3,-1,5),(2,4,6)]` into an upper triangular matrix by using suitable row transformations.

Answer

To transform the matrix \(\begin{pmatrix} 1 & 2 & 4 \\ 3 & -1 & 5 \\ 2 & 4 & 6 \end{pmatrix}\) into an upper triangular matrix using suitable row transformations, we will follow these steps:

### Step 1: Write down the original matrix
\[ A = \begin{pmatrix} 1 & 2 & 4 \\ 3 & -1 & 5 \\ 2 & 4 & 6 \end{pmatrix} \]

### Step 2: Eliminate the first column below the first row
We will perform row operations to make the elements below the first row in the first column equal to zero.

- For Row 2: \( R_2 = R_2 - 3R_1 \)
- For Row 3: \( R_3 = R_3 - 2R_1 \)

Calculating these:
- \( R_2 = (3, -1, 5) - 3(1, 2, 4) = (3 - 3, -1 - 6, 5 - 12) = (0, -7, -7) \)
- \( R_3 = (2, 4, 6) - 2(1, 2, 4) = (2 - 2, 4 - 4, 6 - 8) = (0, 0, -2) \)

Now, the matrix looks like:
\[ \begin{pmatrix} 1 & 2 & 4 \\ 0 & -7 & -7 \\ 0 & 0 & -2 \end{pmatrix} \]

### Step 3: Make the leading coefficient of Row 2 positive
To make the leading coefficient of Row 2 positive, we can multiply Row 2 by -1:
\[ R_2 = -1 \cdot R_2 = -1 \cdot (0, -7, -7) = (0, 7, 7) \]

Now the matrix becomes:
\[ \begin{pmatrix} 1 & 2 & 4 \\ 0 & 7 & 7 \\ 0 & 0 & -2 \end{pmatrix} \]

### Step 4: Make the leading coefficient of Row 3 positive
To make the leading coefficient of Row 3 positive, we can multiply Row 3 by -1:
\[ R_3 = -1 \cdot R_3 = -1 \cdot (0, 0, -2) = (0, 0, 2) \]

Now the matrix becomes:
\[ \begin{pmatrix} 1 & 2 & 4 \\ 0 & 7 & 7 \\ 0 & 0 & 2 \end{pmatrix} \]

### Final Result
The final upper triangular matrix is:
\[ \begin{pmatrix} 1 & 2 & 4 \\ 0 & 7 & 7 \\ 0 & 0 & 2 \end{pmatrix} \]