GATE 2023 EE Q.24: solution and answer

GATE 2023 · EEQ.24 · 1 mark · MCQ

In the figure, the vectors $u$ and $v$ are related as: $Au = v$ by a transformation matrix $A$ . The correct choice of $A$ is

GATE 2023 EE question 24: Two arrows drawn from a common tail at the origin, labelled (0, 0). — Figure from the GATE 2023 EE paper. It is part of the question.

(A)
$[\frac{4}{5} - \frac{3}{5} \frac{3}{5} \frac{4}{5}]$
(B)
$[\frac{4}{5} \frac{3}{5} - \frac{3}{5} \frac{4}{5}]$
(C)
$[\frac{4}{5} \frac{3}{5} \frac{3}{5} \frac{4}{5}]$
(D)
$[\frac{4}{5} \frac{3}{5} - \frac{3}{5} - \frac{4}{5}]$

Answer and solution

ANSWER

The correct choice is (A), $A = [\frac{4}{5} - \frac{3}{5} \frac{3}{5} \frac{4}{5}]$ . It is the only option with $A [43] = [50]$ . Geometrically, it is the clockwise rotation by $θ = tan^{- 1} (3/4)$ .

Official answer: ASource: Official answer key, GATE 2023 EE (IIT Kanpur)

From the figure, both vectors start at the origin, so $u = [43]$ and $v = [50]$ .
The single condition $Au = v$ gives only two equations for the four entries of $A$ , so it does not fix $A$ on its own. What we can settle is which of the four given options satisfies it. The reliable method is to multiply each option by $u$ .
Option (A): $[\frac{4}{5} - \frac{3}{5} \frac{3}{5} \frac{4}{5}] [43] = [\frac{16}{5} + \frac{9}{5} - \frac{12}{5} + \frac{12}{5}] = [50] = v$ So (A) works.
The other options fail. (B) gives $[\frac{16}{5} - \frac{9}{5} \frac{12}{5} + \frac{12}{5}] = [\frac{7}{5} \frac{24}{5}]$ . (C) gives $[\frac{16}{5} + \frac{9}{5} \frac{12}{5} + \frac{12}{5}] = [5 \frac{24}{5}]$ . (D) gives $[\frac{16}{5} - \frac{9}{5} \frac{12}{5} - \frac{12}{5}] = [\frac{7}{5} 0]$ . Its second component is 0, but its first is not 5. None of these equals $v$ , so (A) is the only correct option.
Why (A) makes geometric sense: $∣ u ∣ = 4^{2} + 3^{2} = 5 = ∣ v ∣$ , and $u$ makes an angle $θ$ above the x-axis with $cos θ = \frac{4}{5}$ and $sin θ = \frac{3}{5}$ . A clockwise rotation by $θ$ brings $u$ down onto the x-axis without changing its length. That rotation is $R (- θ) = [cos θ - sin θ sin θ cos θ]$ , which is exactly matrix (A).
Matrix (B) is $R (θ)$ , the anticlockwise rotation. It turns $u$ further up, to angle $2 θ$ : $5 (cos 2 θ, sin 2 θ) = 5 (\frac{16}{25} - \frac{9}{25}, \frac{2 \cdot 3 \cdot 4}{25}) = (\frac{7}{5}, \frac{24}{5})$ . This agrees with the product in step 4. Answer: (A).

KEY IDEA

With four options to choose from, check $Au = v$ by multiplying each option by $u$ . As a sanity check: $u$ and $v$ are both 5 long, and turning $u$ clockwise by $θ$ (with $cos θ = 4/5$ , $sin θ = 3/5$ ) lands it on $v$ . That rotation is $R (- θ) = [cos θ - sin θ sin θ cos θ]$ .

WATCH OUT FOR

Picking (B) because it looks like the textbook rotation matrix $[cos θ sin θ - sin θ cos θ]$ . That matrix rotates anticlockwise, but $u$ has to turn clockwise to reach $v$ . (B) actually sends $u$ to $(7/5, 24/5)$ . A second trap is checking only that the y-component comes out 0: (D) passes that test but gives $(7/5, 0)$ , not $(5, 0)$ .

From the GATE 2023 EE paper. Every question of this paper · Download the full paper · All 46 Linear algebra questions, with notes

GATE 2023 EE Q.24: solution and answer

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