Suppose that the n×n matrices A and B​ commute; that​ is, that AB=BA. Prove that eA+B=eAeB.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Question
Suppose that the
n×n
matrices A and B​ commute; that​ is, that
AB=BA.
Prove that
eA+B=eAeB.
 
 
 
Expand the right hand side
eAeB
using the fact that if A is an
n×n
​matrix, then the exponential matrix
eA
is the
n×n
matrix defined by the series
eA=
 
Bold Upper A plus StartFraction Bold Upper A squared Over 2 exclamation mark EndFraction plus StartFraction Bold Upper A cubed Over 3 exclamation mark EndFraction midline ellipsis plus StartFraction Bold Upper A Superscript n Over n exclamation mark EndFraction plus midline ellipsis .A+A22!+A33!⋯+Ann!+⋯.
Bold Upper A plus StartFraction Bold Upper A squared Over 2 EndFraction plus StartFraction Bold Upper A cubed Over 3 EndFraction midline ellipsis plus StartFraction Bold Upper A Superscript n Over n EndFraction plus midline ellipsis .A+A22+A33⋯+Ann+⋯.
Bold Upper I plus Bold Upper A plus StartFraction Bold Upper A squared Over 2 exclamation mark EndFraction plus midline ellipsis plus StartFraction Bold Upper A Superscript n Over n exclamation mark EndFraction plus midline ellipsis .I+A+A22!+⋯+Ann!+⋯.
Bold Upper I plus Bold Upper A plus StartFraction Bold Upper A squared Over 2 EndFraction plus midline ellipsis plus StartFraction Bold Upper A Superscript n Over n EndFraction plus midline ellipsis .
Expert Solution
steps

Step by step

Solved in 2 steps with 1 images

Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,