## Acalytica

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Let $A$ be a square matrix. Proof or Counterexample.
a) If $A$ is diagonalizable, then so is $A^{2}$.
b) If $A^{2}$ is diagonalizable, then so is $A$.
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a) Proof: If a square matrix $A$ is diagonalizable, then there exists an invertible matrix $P$ and a diagonal matrix $D$ such that $A=P D P^{-1}$. Hence, $A^2 = PDP^{-1}PDP^{-1} = PD^2P^{-1}$ where $D^2$ is a diagonal matrix as well. Hence, $A^2$ is diagonalizable.
b) Counterexample: Consider the matrix
$A = \begin{bmatrix}0 & 1 \\ 0 & 0\end{bmatrix}$
It follows that $A^2=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]$, which is diagonalizable. However, the matrix $A$ is not diagonalizable, as its only eigenvalue is 0 , and it is not invertible. Hence, the statement is false.
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