Find the incorrectness of the process.

Question

Assume $$A\xi=\lambda_1\xi $$ then $$A\xi\xi^T=\lambda_1\xi\xi^T $$ and $$|A\xi\xi^T|=|\lambda_1\xi\xi^T| $$ which we will have $$|A|\cdot |\xi\xi^T|=\lambda_1^n\cdot|\xi\xi^T|$$ then $$|A|=\lambda_1^n$$ But $$|A|=\prod_{i=1}^{n}\lambda_i$$

What's the problem in this process?

Answer 1

Hint: Take a good look at $|\xi\xi^T|$. Think about, say, the rank of that matrix and what that means for the determinant.