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GATE 2023 EE – Question 24

Engineering Mathematics · Linear Algebra: Matrix Algebra · 1 mark · Multiple choice

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

(In the figure, $\mathbf u$ goes from (0,0) to (4,3) and $\mathbf v$ from (0,0) to (5,0).)

Diagram for GATE 2023 EE question 24
  1. $\begin{bmatrix}\frac45&\frac35\\-\frac35&\frac45\end{bmatrix}$
  2. $\begin{bmatrix}\frac45&-\frac35\\\frac35&\frac45\end{bmatrix}$
  3. $\begin{bmatrix}\frac45&\frac35\\\frac35&\frac45\end{bmatrix}$
  4. $\begin{bmatrix}\frac45&-\frac35\\\frac35&-\frac45\end{bmatrix}$

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Show answer and explanation

Correct answer: (A) $\begin{bmatrix}\frac45&\frac35\\-\frac35&\frac45\end{bmatrix}$

Explanation

Both vectors have length 5, so $\mathbf A$ is a rotation taking $(4,3)$ onto $(5,0)$, i.e. a clockwise rotation by $\theta$ with $\cos\theta=\frac45$, $\sin\theta=\frac35$: $\mathbf A=\begin{bmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{bmatrix}$. Check: $\left(\frac{16}5+\frac95,\ -\frac{12}5+\frac{12}5\right)=(5,0)$ ✓.