Let A,B,C be n times n matrices. If B=CAC^{-1} show that det(A)=det(B)

Ava-May Nelson 2020-11-09 Answered
Let A,B,C be n×n matrices. If B=CAC1 show that det(A)=det(B)
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

diskusje5
Answered 2020-11-10 Author has 82 answers
Step 1
Consider that A, B, C are n×n matrices.
Given: B=CAC1
To Show: det(A)=det(B)
Apply property of determinants which says if A1,A2,A3, …An are n×n matrices. Then det(A1A2A3An)=det(A1)det(A2)det(A3)det(An). Therefore
det(B)=det(CAC1)
=det(C)det(A)det(C1)
Step 2
Apply commutative laws of determinants that is, det(A1)det(A2)=det(A2)det(A1).
det(B)=det(CAC1)
=det(C)det(A)det(C1)
=det(C)det(C1)det(A)
Apply property of determinants that det(A11)=1det(A1)
det(B)=det(CAC1)
=det(C)det(A)det(C1)
=det(C)det(C1)det(A)
=det(C)1det(C)det(A)
=det(A)
Not exactly what you’re looking for?
Ask My Question
Jeffrey Jordon
Answered 2022-01-22 Author has 2262 answers

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

New questions