Prove that if A is diagonalizable (with diagonal matrix D made up of the eigenva

Question

Prove that if A is diagonalizable (with diagonal matrix D made up of the eigenvalues as entries on the main diagonal), then det(e^A) = etrace(D) where the trace of a matrix A is the sum of the main diagonal elements. (Note: This is true for all square matrices.)

12) Prove that if A is diagonalizable (with diagonal matrix D made up of the eigenvalues as entries on the main diagonal), then det(eA) etrace(D) where the trace of a matrix A is the sum of the main diagonal elements. (Note: This is true for all square matrices.)

Explanation / Answer

As, A is diagonalizable

A=P(?1)DP

with P as invertible.

Let ?1,…,?n be the eigenvalues of A

Note that Dk is upper-triangular with ?k1,…,?kn on the diagonal. Also eD is upper triangular with e?1,…,e?n on the diagonal. So

det eD=e?1?e?n=e?1+…+?n=etrD

As we know trA=trD and that Dk=PAkP?1 for all k. Therefore,

PeAP?1=eD

? det eA=det(PeAP?1) = det eD=etrD

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