The GATE Grind

GATE 2024 DA – Question 48

Linear Algebra · Systems of linear equations and Gaussian elimination · 2 marks · Multiple select

Which of the following statements is/are TRUE?
Note: $\mathbb{R}$ denotes the set of real numbers.

  1. There exist $M \in \mathbb{R}^{3 \times 3}$, $p \in \mathbb{R}^3$, and $q \in \mathbb{R}^3$ such that $Mx = p$ has a unique solution and $Mx = q$ has infinite solutions.
  2. There exist $M \in \mathbb{R}^{3 \times 3}$, $p \in \mathbb{R}^3$, and $q \in \mathbb{R}^3$ such that $Mx = p$ has no solutions and $Mx = q$ has infinite solutions.
  3. There exist $M \in \mathbb{R}^{2 \times 3}$, $p \in \mathbb{R}^2$, and $q \in \mathbb{R}^2$ such that $Mx = p$ has a unique solution and $Mx = q$ has infinite solutions.
  4. There exist $M \in \mathbb{R}^{3 \times 2}$, $p \in \mathbb{R}^3$, and $q \in \mathbb{R}^3$ such that $Mx = p$ has a unique solution and $Mx = q$ has no solutions.

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Show answer and explanation

Correct answer: (B) There exist $M \in \mathbb{R}^{3 \times 3}$, $p \in \mathbb{R}^3$, and $q \in \mathbb{R}^3$ such that $Mx = p$ has no solutions and $Mx = q$ has infinite solutions.; (D) There exist $M \in \mathbb{R}^{3 \times 2}$, $p \in \mathbb{R}^3$, and $q \in \mathbb{R}^3$ such that $Mx = p$ has a unique solution and $Mx = q$ has no solutions.

Explanation

A: a unique solution for a square matrix means that $M$ is invertible, and then $Mx = q$ has exactly one solution for every $q$, so A is false. B: a singular $M$ has no solution for $p$ outside its column space and infinitely many for $q$ inside it, so B is true. C: a $2 \times 3$ matrix has rank at most 2, so it has a free variable and can never give a unique solution, so C is false. D: a $3 \times 2$ matrix of rank 2 gives a unique solution when $p$ is in its column space and no solution when $q$ is outside it, so D is true.