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- Jan 26, 2012

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**Problem**: Let $A$ be a $n \times n$ matrix with real entries. Prove that if $A$ is symmetric, that is $A = A^T$ then all eigenvalues of $A$ are real.

**Solution**: I'm definitely not seeing how to approach this problem. I know that to calculate the eigenvalues of a matrix I need to solve $\text{det }(A-\lambda I)=0$ and I have experience calculating them, but I've never seen commentary on whether the values will be real or complex. Any ideas to get started?

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