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Similarity implies equality?

Let $A\in \mathbb{C}^{n\times n}$ be similar to $A^2$ . Does $A=A^2$ hold?

Solution

No! Take $A=\begin{pmatrix}1 &0 \\1&1 \end{pmatrix}$ then $A^2=\begin{pmatrix}1 &0\\ 2&1 \end{pmatrix}$ . The matrices are similar but not equal.

Tags: Linear Algebra

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