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

Question
Matrices
asked 2020-11-11
Prove that if A and B are similar n x n matrices, then tr(A) = tr(B).

Answers (1)

2020-11-12
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 \(B=P^{-1}AP \text{ and } A=PBP^{-1}\)
Step 2
Now
tr(B)=tr(P^{-1}AP)\)
\(= tr(P^{-1}PA) \left[ \because \text{If A,B and C are any matrices } A(B \cdot C)=(AB)C , \text{ provided both sides are defined} \right]\)
\(=tr(IA) , \text{ where I is the identity matrix}\)
\(=tr(A) \left[ \because \text{ for any } n \times n \text{matrix} A,I \cdot A =A \right]
0

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