Answer
The $5\times7$ matrix must have five pivot columns.
Work Step by Step
Suppose there were only four pivot columns. Then the matrix equation $\mathbf{A}\vec{x}=\vec{0}$ would have four basic variables and three free variables, i.e., it would have four linearly independent columns. But four linearly independent vectors span only a 4-dimensional subspace of $\mathbb{R}^{5}$. Hence, the matrix must have more than four pivot columns. But a matrix cannot have more pivot positions than there are rows, so the matrix must have exactly five pivot columns.
