Show that if A and B are similar matrices, then

verskalksv

verskalksv

Answered question

2021-12-05

Show that if A and B are similar matrices, then det(A)=det(B).

Answer & Explanation

Himin1945

Himin1945

Beginner2021-12-06Added 12 answers

Let A and B be similar. Then there exist a nonsingular matrix S such that
A=S1BS.
We have that
det(A)=det(S1BS)
=det(S1)det(B)det(S)
=det(S1)det(S)det(B)
=det(S1S)det(B)
=det(I)det(B)=det(B).
Result:
Since A and B are similar, there exist a nonsigular S such that A=S1BS. Then, using the fact that det(AB)=det(A)det(B) for any matrices A and B, the result is easily obtained.

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