<s>[INST] The pivot column of a matrix $A$ is a column that contains a pivot position, and the pivot position is the position of an element in a row echelon form that is the first nonzero entry in its row and whose column contains all zeros elsewhere, so that the position is the location of a leading entry of a $1$ (or in general a leading entry of a $1$ or $-1$) in row echelon form for $A$. For a $4\times 5$ matrix $A$, find the smallest number $k$ such that the first $k$ columns of $A$ have $4$ pivot columns.[/INST]