031 Review Part 1, Problem 2

True or false: If a matrix  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A^{2}}   is diagonalizable, then the matrix  $A$  must be diagonalizable as well.

Solution:
Let  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A={\begin{bmatrix}0&1\\0&0\end{bmatrix}}.}
First, notice that
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A^{2}={\begin{bmatrix}0&0\\0&0\end{bmatrix}},}
which is diagonalizable.
Since  $A$  is a diagonal matrix, the eigenvalues of  $A$  are the entries on the diagonal.
Therefore, the only eigenvalue of  $A$  is  $0.$  Additionally, there is only one linearly independent eigenvector.
Hence,  $A$  is not diagonalizable and the statement is false.

Final Answer:
FALSE

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