Is det(A^n) equal to det(A)^n for all matrices A?

Thread starter Hemmer
Start date Oct 24, 2007

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The discussion centers on proving that det(A^n) equals det(A)^n for all matrices A. The initial attempt uses the property det(AB) = det(A)det(B) and applies it to show that det(A^2) = det(A)^2. However, participants note that this is insufficient for a complete proof. The suggestion of using mathematical induction is proposed as a more effective method to establish the equality for all n. The conversation emphasizes the need for a rigorous proof beyond the initial observations.

Oct 24, 2007

Hemmer

Question. Prove det(A^n) = det(A) ^ n

I have:

det(AB) = det(A)det(B) (holds for any number of factors)
set A=B
det(AA) = det(A)det(A)
det(A^2) = det(A)^2 ?

Is this enough for a proof?

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Oct 24, 2007

radou

Homework Helper

Hemmer said:

Is this enough for a proof?

Of course not, since you didn't prove what you need to prove. Hint: induction.

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