Prove that if A and B are similar n x n matrices, then tr(A) = tr(B).

sagnuhh 2020-11-11 Answered
Prove that if A and B are similar n x n matrices, then tr(A) = tr(B).
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AGRFTr
Answered 2020-11-12 Author has 95 answers

Step 1
Given that A and B are similar n x n matrices.
Then there exist a non-singular n x n matrix P such that B=P1AP and A=PBP1
Step 2
Now
tr(B)=tr(P1AP))
=tr(P1PA)[If A,B and C are any matrices A(BC)=(AB)C, provided both sides are defined]
=tr(IA), where I is the identity matrix
=tr(A)[ for any n×n matrixA,IA=A]

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Jeffrey Jordon
Answered 2022-01-22 Author has 2313 answers

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