Matrix similarity theorem(matrix representation)

[cold totem]

Hello, I'm new to this similarity topic so I'm wondering why 2 similar matrices mean that the expressions of C - lambda x I and B - lambda x I are singular. How would I go about proving it? Any advice would be much appreciated

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[ionic mist]

Hello

I think here a straightforward proof using the definition of similarity would suffice

Just remember that if you have an invertible matrix P, you can write lambda * id as lambda * P^-1 * P