GATE CS 2021 Set 2 Q.34 (CS Q.24): Linear Algebra

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 ______.

Your answer

Official answer and solution

Official answer: 4
From: Official GATE 2021 answer key

Solution

Use the Rank-Nullity Theorem by relating the dimension of the solution space of the homogeneous linear system to the matrix rank.

The matrix $P$ has dimensions $4 \times 5$ , so the number of columns (the dimension of the domain) is $n = 5$ .

The set of all solutions to the homogeneous equation $Px = 0$ is the null space (kernel) of $P$ , denoted $\text{Null}(P)$ .

We are given that every solution of $Px = 0$ is a scalar multiple of the single non-zero vector $[2\; 5\; 4\; 3\; 1]^T$ . This means the null space is spanned by one non-zero vector, so its dimension (nullity) is:

\text{nullity}(P) = \dim(\text{Null}(P)) = 1

By the Rank-Nullity Theorem:

\text{rank}(P) + \text{nullity}(P) = n

\text{rank}(P) + 1 = 5 \implies \text{rank}(P) = 4

Final answer

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2021 Computer Science and Information Technology - Session 2 (CS2)
Question: 34 of 65, printed as CS Q.24
Section: Computer Science and Information Technology

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