The given matrix is:

Perform the following row operations:
$R_2 \leftarrow R_2 - 5R_1$
$R_3 \leftarrow R_3 - 2R_1$
$R_4 \leftarrow R_4 - 3R_1$

The matrix becomes:

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

Perform the row operation:
$R_2 \leftarrow \frac{1}{4}R_2$

The matrix becomes:

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

Perform the following row operations:
$R_3 \leftarrow R_3 - 5R_2$
$R_4 \leftarrow R_4 - 7R_2$

The matrix becomes:

$$\begin{bmatrix} 1 & 0 & 3 & 0 \\ 0 & 1 & -\frac{13}{4} & \frac{1}{2} \\ 0 & 0 & \frac{45}{4} & -\frac{5}{2} \\ 0 & 0 & \frac{75}{4} & -\frac{3}{2} \end{bmatrix}$$

Perform the row operation:
$R_3 \leftarrow \frac{4}{45}R_3$

The matrix becomes:

$$\begin{bmatrix} 1 & 0 & 3 & 0 \\ 0 & 1 & -\frac{13}{4} & \frac{1}{2} \\ 0 & 0 & 1 & -\frac{2}{9} \\ 0 & 0 & \frac{75}{4} & -\frac{3}{2} \end{bmatrix}$$

Perform the row operation:
$R_4 \leftarrow R_4 - \frac{75}{4}R_3$

The matrix becomes: