$A= \begin{bmatrix} 1 & -1 & 3 & 6 \[0.3em] 1 & 3 &-3 &-4 \[0.3em] 5 & 3 & 3 &11 \end{bmatrix} $   * By using R2 - R1; R3 - 5R1; $A = \begin{bmatrix} 1 & -1 & 3 &6 \[0.3em] 0 & 4 & -6 &-10\[0.3em] 0 & 8 & -12 &-19 \end{bmatrix}$   * By using C2 + C1; C3 - 3C1; C4 - 6C1; $A = \begin{bmatrix} 1 & 0 & 0 &0 \[0.3em] 0 & 4 & -6 &-10\[0.3em] 0 & 8 & -12 &-19 \end{bmatrix}$   * By using$ \dfrac1 4 C_2 ; -\dfrac1 6C_3 ;$ $A = \begin{bmatrix} 1 & 0 & 0 &0 \[0.3em] 0 & 1 & 1 &-10\[0.3em] 0 & 2 & 2 &-19 \end{bmatrix}$   * By using$R_3 - 2R_2 ;$ $A = \begin{bmatrix} 1 & 0 & 0 &0 \[0.3em] 0 & 1 & 1 &-10\[0.3em] 0 & 0 & 0 &1 \end{bmatrix}$   * By using$C_3 - C_2 ; C_4+10 C_2;$ $A = \begin{bmatrix} 1 & 0 & 0 &0 \[0.3em] 0 & 1 & 0 &0\[0.3em] 0 & 0 & 0 &1 \end{bmatrix}$   * By using$C_3 \leftrightarrow C_4$ $A = \begin{bmatrix} 1 & 0 & 0 &0 \[0.3em] 0 & 1 & 0 &0\[0.3em] 0 & 0 & 1 &0 \end{bmatrix}$

which is the required normal form.

• Hence Rank=Number of non-zero rows = 3