To transform the given matrix into echelon form, we will perform a series of elementary row operations.

To transform the given matrix into echelon form, we will perform a series of elementary row operations.

The given matrix is:

$$\begin{bmatrix} 1 & 0 & 3 & 0 \\ 5 & 4 & 2 & 2 \\ 2 & 5 & 1 & 0 \\ 3 & 7 & 5 & 2 \end{bmatrix}$$

Step 1: Make the entries below the leading 1 in the first column zero.
Perform the following row operations:
$R_2 \leftarrow R_2 - 5R_1$
$R_3 \leftarrow R_3 - 2R_1$
$R_4 \leftarrow R_4 - 3R_1$

Calculations for the new rows:
$R_2$: $[5, 4, 2, 2] - 5[1, 0, 3, 0] = [5-5, 4-0, 2-15, 2-0] = [0, 4, -13, 2]$
$R_3$: $[2, 5, 1, 0] - 2[1, 0, 3, 0] = [2-2, 5-0, 1-6, 0-0] = [0, 5, -5, 0]$
$R_4$: $[3, 7, 5, 2] - 3[1, 0, 3, 0] = [3-3, 7-0, 5-9, 2-0] = [0, 7, -4, 2]$

The matrix becomes:

\begin{bmatrix} 1 & 0 & 3 & 0 \\ 0 & 4 & -13 & 2 \\ 0 & 5 & -5 & 0 \\ 0 & 7 & -4 &