Suppose that $P$ is a $4 \times 5$ matrix such that every solution of the equation $Px = 0$ is a scalar multiple of $[2\; 5\; 4\; 3\; 1]^T$. The rank of $P$ is _______.

According to the Rank-Nullity Theorem, for any matrix $P$ with $n$ columns:

Here, $P$ is a $4 \times 5$ matrix, so the number of columns is $n = 5$.

The null space of $P$ is the set of all vectors $x$ satisfying $Px = 0$. Since every solution is a scalar multiple of the single non-zero vector $[2\; 5\; 4\; 3\; 1]^T$, the null space has dimension $1$:

Applying the Rank-Nullity Theorem:

Thus, the rank of $P$ is 4.