GATE 2024 DA Q.48: solution and answer

GATE 2024 · DAQ.48 · 2 marks · MSQ

Which of the following statements is/are TRUE?

Note: $R$ denotes the set of real numbers.

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

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Answer and solution

ANSWER

Statements (B) and (D) are true.

Official answer: B, DSource: Official answer key, GATE 2024 DA (IISc Bengaluru)

Recall: if $Mx = b$ is consistent, its solution is unique exactly when the columns of $M$ are independent (nullity $0$ ). If it is consistent and the nullity is positive, there are infinitely many solutions. There is no solution when $b$ lies outside the column space of $M$ .
(A) A unique solution of a square $3 \times 3$ system means nullity $0$ , so $rank M = 3$ and $M$ is invertible. Then $Mx = q$ has exactly one solution for every $q$ , never infinitely many. False.
(B) Take $M = 100000000$ , $p = 010$ , $q = 000$ . The second row of $Mx = p$ reads $0 = 1$ , so there is no solution. For $q = 0$ the system is consistent and the nullity is $3 - 1 = 2 > 0$ , so there are infinitely many solutions. True.
(C) For $M \in R^{2 \times 3}$ , $rank M \leq 2$ , so by rank-nullity the nullity is $3 - rank M \geq 1$ . If $Mx = p$ has a solution, adding any non-zero null-space vector gives another, so the solution is never unique. False.
(D) Take $M = 100010$ , whose columns are independent (nullity $0$ ). With $p = 110$ , $Mx = p$ gives $x_{1} = 1$ , $x_{2} = 1$ , and the third row $0 = 0$ holds, so the solution is unique. With $q = 001$ the third row reads $0 = 1$ , so there is no solution. True.
Hence the true statements are (B) and (D).

KEY IDEA

Uniqueness of a solution depends only on the nullity of $M$ (independent columns), while existence depends on whether the right-hand side lies in the column space. So the same $M$ can give different outcomes for different right-hand sides only when the nullity is positive (infinite versus none) or when the system is tall (unique versus none).

WATCH OUT FOR

Thinking that one matrix cannot give different outcomes for different right-hand sides. For a square invertible matrix every right-hand side behaves alike, but a singular square matrix (B) or a tall matrix with independent columns (D) can have a solution for one vector and none for another. In (C), forgetting that a wide matrix always has a non-trivial null space.

From the GATE 2024 DA paper. Every question of this paper · Download the full paper · All 2 Linear systems questions, with notes