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Which one of the following statements is NOT true for a square matrix $A$?

- If $A$ is upper triangular, the eigenvalues of $A$ are the diagonal elements of it
- If $A$ is real symmetric, the eigenvalues of $A$ are always real and positive
- If $A$ is real, the eigenvalues of $A$ and $A^{T}$ are always the same
- If all the principal minors of $A$ are positive, all the eigenvalues of $A$ are also positive