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Show that similar matrices have same characteristic polynomial.
asked Nov 11, 2017 by randomisation (1,920 points)  

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The similar matrix of a matrix $A$ is $P^{-1}AP$ where P is invertible.

The characteristic equation of $P^{-1}AP$ is $|P^{-1}AP - \lambda I| = 0$. Now,

$|P^{-1}AP - \lambda I| \\ = |P^{-1}AP - P^{-1}\lambda IP| \\ = |P^{-1}(A-\lambda I)P \\ = |P^{-1}||A-\lambda I||P| \\ = |A-\lambda I| = 0$

which is the characteristic equation of the matrix $A$. Hence, proved.
answered Nov 11, 2017 by randomisation (1,920 points)  
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