Which of the following statements is/are TRUE? Note: R denotes the set of real numbers.

There exist M ∈ R^3×3, p ∈ R^3, and q ∈ R^3 such that Mx = p has no solutions and Mx = q has infinite solutions.

There exist M ∈ R^3×2, p ∈ R^3, and q ∈ R^3 such that Mx = p has a unique solution and Mx = q has no solutions.

There exist M ∈ R^3×3, p ∈ R^3, and q ∈ R^3 such that Mx = p has a unique solution and Mx = q has infinite solutions.

There exist M ∈ R^3×3, p ∈ R^3, and q ∈ R^3 such that Mx = p has no solutions and Mx = q has infinite solutions.

There exist M ∈ R^2×3, p ∈ R^2, and q ∈ R^2 such that Mx = p has a unique solution and Mx = q has infinite solutions.

There exist M ∈ R^3×2, p ∈ R^3, and q ∈ R^3 such that Mx = p has a unique solution and Mx = q has no solutions.

GATE DA 2024 Q48: Possible solution counts of Mx = p and Mx = q

Linear Algebra
2 marks
MSQ
No negative marking

Which of the following statements is/are TRUE?

Note: $\mathbb{R}$ denotes the set of real numbers.

Try it: multiple select: one or more correct options

Select all correct options. No negative marking, no partial marks.

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Answer: B, D (official GATE 2024 final answer key)

Concept used: Uniqueness depends only on the null space of $\mathbf{M}$ ; existence depends on whether the right side is in the column space.

First step: A unique solution means the null space is $\{\mathbf{0}\}$ , so no right-hand side can then have infinitely many; a $2\times3$ matrix always has a non-trivial null space.

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Source: official GATE 2024 DA question paper and final answer key, organised by IISc Bangalore. Question paper · Answer key

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GATE DA 2024 Q48: Possible solution counts of Mx = p and Mx = q

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