Step 1:

R1 ⇔ R2

A = $\left[\begin{array}{cccc} 1 & 0 & 4 & 3 \\ 0 & 1 & -3 & -1 \\ 3 & 1 & 0& 2 \\ 1 & 1 & -2 & 0 \end{array}\right]$

Step 2:

R3 ⇒ R3 - 3(R1)

R4 ⇒ R4 - R1

A = $\left[\begin{array}{cccc} 1 & 0 & 4 & 3 \\ 0 & 1 & -3 & -1 \\ 0 & 1 & -12 & -7 \\ 0 & 1 & -6 & -3 \end{array}\right]$

Step 3:

C3 ⇒ C3 - 4(C1)

C4 ⇒ C4 - 3(C1)

A = $\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & -3 & -1 \\ 0 & 1 & -12 & -7 \\ 0 & 1 & -6 & -3 \end{array}\right]$

Step 4:

R3 ⇒ R3 - R2

R4 ⇒ R4 - R2

A = $\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & -3 & -1 \\ 0 & 0 & -9 & -6 \\ 0 & 0 & -3 & -2 \end{array}\right]$

Step 5:

C3 ⇒ C3 + 3(C2)

C4 ⇒ C4 + C2

A = $\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -9 & -6 \\ 0 & 0 & -3 & -2 \end{array}\right]$

Step 6:

R3 ⇒ R3 + 3(R4)

A = $\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & -3 & -2 \end{array}\right]$

Step 7:

C3 ⇒ C3 / (-3)

A = $\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & -2 \end{array}\right]$

Step 8:

C4 ⇒ C4 + 2(C3)

A = $\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]$

Step 9:

R3 ⇔ R4

A = $\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

Rank of matrix A = number of non zero rows

i.e. Rank of matrix A = 3