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No, a general $n \times n$ matrix need not have $n$ distinct real eigenvalues. Some matrices may have repeated eigenvalues, complex eigenvalues, or fewer than $n$ linearly independent eigenvectors.

Conditions: The matrix is $n \times n$.; The field of scalars is typically $\mathbb{R}$ unless specified otherwise.

No, a general $n \times n$ matrix need not have $n$ distinct real eigenvalues. Some matrices may have repeated eigenvalues, complex eigenvalues, or fewer than $n$ linearly independent eigenvectors.

Conditions: The matrix is $n \times n$.; The field of scalars is typically $\mathbb{R}$ unless specified otherwise.