$If A is an nxn matrix, then A has n distinct eigenvalues.True or false?$

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$If A is an nxn matrix, then A has n distinct eigenvalues.True or false?$

Need help with explaining this. If true, give a reason.If false, give a counterexample.

Matrices Eigenvalues & Eigenvectors

Lol Cheng

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The statement is false. As a counterexample, let
$$A=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}=I. $$
Let $\lambda$ be an eigenvalue. Then
\[\text{det}(A-\lambda I)=\text{det}\begin{pmatrix}1-\lambda & 0 \\ 0 & 1-\lambda \end{pmatrix} \]
\[=(1-\lambda)^2=0.\]
Hence
\[\lambda=1,1.\]
So there is only one eigenvalue and NOT two distinct eigenvalues.

Erdos

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Erdos

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$If A is an nxn matrix, then A has n distinct eigenvalues.True or false?$

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