let take a 3x3 matrix instead of 4x4 matrix for simplicity
The rank of a matrix is defined as the maximum number of linearly independent row/ col vectors in the matrix
Here it's given that " one row/column is dependent on the others"
which means out of 3 rows , let R1 , R2 rows are linearly independent and R3 is linear dependent
therefore , rank of the matrix B = 2
therefore rank of (A) = 2
Now if A is singular then adj(A) is also a singular
A = $\begin{bmatrix} a & b & c & \\ d & e&f & \\ 0&0 & 0 & \end{bmatrix}$
B = adj(A) =
$\begin{bmatrix} 0 & 0 & 0 & \\ 0 & 0&0 & \\ bf -e c&dc-af & ae -db & \end{bmatrix}$
$\begin{bmatrix} bf -e c & dc-af & ae -db& \\ 0 & 0&0 & \\ 0 & 0& 0 & \end{bmatrix}$ = rank = 1
Now adj(B) is a null matrix
therefore rank = 0