GATE CS 2021 Set 2 Q24 — Linear Algebra: Matrices, Determinants, System of Linear Equations

We are given that $P$ is a $4 \times 5$ matrix. The equation $Px = 0$ represents a homogeneous system of linear equations.

The set of all solutions to $Px = 0$ forms the null space (or kernel) of the matrix $P$.

According to the problem, every solution of $Px = 0$ is a scalar multiple of the single non-zero vector:

This implies that the null space of $P$ is spanned by this single vector $v$. Since $v \neq 0$, the dimension of the null space (known as the nullity of $P$) is:

By the Rank-Nullity Theorem, for any $m \times n$ matrix $P$:

Here, the number of columns $n = 5$. Substituting the values:

Thus, the rank of $P$ is 4.