GATE 2022 EC Q.12: solution and answer

GATE 2022 · ECQ.12 · 1 mark · MCQ

Consider a system of linear equations $A x = b$ , where
$A = [1 - 1 - 2 2 3 - 3], b = [13] .$
This system of equations admits ________.

(A)
a unique solution for $x$
(B)
infinitely many solutions for $x$
(C)
no solutions for $x$
(D)
exactly two solutions for $x$

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Answer and solution

ANSWER

(C) no solutions for $x$ . Row 2 of $A$ is $- 1$ times row 1, but the right-hand sides are $1$ and $3$ , not $1$ and $- 1$ , so $ρ (A) = 1 \neq = 2 = ρ ([A ∣ b])$ .

Official answer: CSource: Official answer key, GATE 2022 EC (IIT Kharagpur)

Write the augmented matrix: $[A ∣ b] = [1 - 1 - 2 2 3 - 3 13]$
Row reduce with $R_{2} \to R_{2} + R_{1}$ : $[10 - 2 0 3014]$
The second row now reads $0 \cdot x_{1} + 0 \cdot x_{2} + 0 \cdot x_{3} = 4$ , i.e. $0 = 4$ . No $x$ satisfies it. (You get the same thing by adding the two original equations: the left sides cancel and the right sides add to $1 + 3 = 4$ .)
Ranks: row 2 of $A$ is $- 1$ times row 1, and row 1 is non-zero, so $ρ (A) = 1$ . The reduced augmented matrix has two non-zero rows, one of them $[0004]$ , so $ρ ([A ∣ b]) = 2$ .
Consistency (Rouché–Capelli) theorem: $A x = b$ has a solution if and only if $ρ (A) = ρ ([A ∣ b])$ . Here $1 \neq = 2$ , so the system is inconsistent and has no solution. Answer: (C).
Option (D) can never be right for a linear system: the number of solutions is always $0$ , $1$ or infinitely many. If $x_{1}$ and $x_{2}$ were two different solutions, every point $x_{1} + t (x_{2} - x_{1})$ would also be a solution.

KEY IDEA

$A x = b$ is consistent exactly when $ρ (A) = ρ ([A ∣ b])$ . When the coefficient rows are dependent (here $R_{2} = - R_{1}$ ), the right-hand sides must follow the same relation ( $b_{2} = - b_{1}$ ). If they do, there are infinitely many solutions. If they don't, as here ( $3 \neq = - 1$ ), there are none.

WATCH OUT FOR

Seeing 3 unknowns and only 2 equations and answering (B) 'infinitely many solutions' straight away. Having fewer equations than unknowns rules out a unique solution, but it does not make the system consistent. You still have to compare $ρ (A)$ with $ρ ([A ∣ b])$ . Another mistake is to check only $ρ (A)$ and forget the augmented column.

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GATE 2022 EC Q.12: solution and answer

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