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ISI2019-MMA-23

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Let $A$ be $2 \times 2$ matrix with real entries. Now consider the function $f_A(x)$ = $Ax$ . If the image of every circle under $f_A$ is a circle of the same radius, then 

  1. A must be an orthogonal matrix
  2. A must be a symmetric matrix
  3. A must be a skew-symmetric matrix
  4. None of the above must necessarily hold 

2 Answers

Best answer
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Geometric Solution : See A as a transformation matrix.
When the image of every circle is a circle of same radius, it means that there is no scaling or  change in orientation of axes. This occurs when A is a rotation matrix A = $\begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}$ which is an orthogonal matrix.
So $(A)$ is correct
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Don't think complex geometric Transformations. Just analyze : 

"mapping a circle to a circle of the same radius" mean ?? :  Rotating it OR Reflecting it (flipping it over) Must give the same shape. 

Now, this is the Fundamental definition of Orthogonal Matrix. Option A is the correct answer.

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