Suppose that P is a 4 × 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 ________.

GATE CSE 2021 Set 2 Q34: Rank of a 4×5 matrix with a one-dimensional null space

Engineering Mathematics
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Suppose that $P$ is a $4 \times 5$ matrix such that every solution of the equation $P\mathbf{x} = \mathbf{0}$ is a scalar multiple of $[2\;\; 5\;\; 4\;\; 3\;\; 1]^T$ . The rank of $P$ is ________.

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Answer: 4 (official GATE 2021 final answer key)

Concept used: Rank–nullity theorem: rank + nullity = number of columns.

First step: The null space is spanned by one nonzero vector, so its dimension is 1; there are 5 columns.

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GATE CSE 2021 Set 2 Q34: Rank of a 4×5 matrix with a one-dimensional null space

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