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$A \colon\Bbb R^3\to\Bbb R^3$ is a linear transformation which maps the unit sphere to itself. Then $A$ is

a) symmetric;

b) orthogonal;

c) positive definite;

d) symmetric and positive definite.

By the given condition my intuition says $\|Ax\|=\|x\|$ so $A$ will be orthogonal transformation. it may have negative eigenvalue so it will not be positive definite. Its characteristic polynomial will be of degree three and may have a complex root, so in that case it will not be symmetric as symm matricx has eigen value only realss. Thank you for help.

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Hint: Notice that $\displaystyle \left\|A \cdot \frac{x}{\|x\|} \right\|=1$ for all $x \in \mathbb{R}^n\backslash \{0\}$. For the other points, you can think about rotations.

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