To determine if a matrix $A$ has an inverse, we need to check if it is invertible by calculating its determinant. If the determinant is zero, the matrix is not invertible.

For the given matrix:

$A = \begin{bmatrix} 1 & 1 & 1 & 1 \\ 2 & 2 & 2 & 2 \\ 3 & 3 & 3 & 3 \\ 4 & 4 & 4 & 4 \end{bmatrix}$

You can see that each row is a multiple of the first row (e.g., the second row is twice the first row), which means the rows are linearly dependent.

Since the rows are linearly dependent, the determinant is zero, and thus the matrix is singular and does not have an inverse.