Take Class 12 Tuition from the Best Tutors
Affordable fees 1-1 or Group class Flexible Timings Verified Tutors Search in
If A and B are square matrices of the same order such that AB = BA, then prove by induction that
. Further, prove that 
for all n ∈ N
Asked by Shilpa Last Modified
Answer 
Step1
Let (i) P(n):ABn=BnA
Given AB=BA
Put n=1 AB1=B1A.
AB=BA
⇒ P(n) is true for n=1
Step2
To prove P(n) is true for n=k
⇒ABk=BkA
Multiply both sides by B
ABk.B=A(Bk.B)
ABk.B=A.Bk+1[By associative property]
RHS
ABk.B
⇒Bk.AB.
Given AB=BA.
Hence Bk.BA=(Bk.B)A=Bk+1.A
we have LHS=A.Bk+1
Hence LHS=RHS
Hence p(k+1) is true for n=k+1.
⇒ By mathematical induction P(n) is true for n∈N
Step3
(ii)P(n):(AB)n=AnBn.
Let n=1 AB1=A1B1
Given AB=BA
Hence P(n) is true for n=1.
Step4
LHS:
Let P(n) be true for n=k.
Multiply both side by AB
⇒(AB)k=AkBk
(AB)k.AB
⇒AkBk.AB.
⇒(AB)k+1
RHS:
AkBk.AB
AkBk(BA)
Ak(Bk.B)A
Ak.(Bk+1.A)
Ak.ABk+1[ABn=BnA]
Ak+1Bk+1
⇒ P(n) is true for n=k+1.
By principle of mathematical induction
P(n) is true for all n∈N.
read lessRelated Questions
Now ask question in any of the 1000+ Categories, and get Answers from Tutors and Trainers on UrbanPro.com
Ask a QuestionRecommended Articles
Meet Mohammad Wazid, a skilled trainer for...
Mohammad Wazid is a certified professional tutor for class 11 students. He has 6 years of teaching experience which he couples with an energetic attitude and a vision of making any subject easy for the students. Over the years he has developed skills with a capability of understanding the requirements of the students. This...
Read full article >
Meet Raghunandan.G.H, a B. Tech Tutor from...
Raghunandan is a passionate teacher with a decade of teaching experience. Being a skilled trainer with extensive knowledge, he provides high-quality BTech, Class 10 and Class 12 tuition classes. His methods of teaching with real-time examples makes difficult topics simple to understand. He explains every concept in-detail...
Read full article >
Meet Swati, a Hindi Tutor from Bangalore
Swati is a renowned Hindi tutor with 7 years of experience in teaching. She conducts classes for various students ranging from class 6- class 12 and also BA students. Having pursued her education at Madras University where she did her Masters in Hindi, Swati knows her way around students. She believes that each student...
Read full article >
Meet Radhe Shyam Burman, an MBA Tutor from...
Radhe Shyam is a highly skilled accounts and finance trainer with 8 years of experience in teaching. Accounting is challenging for many students and that’s where Radhe Shyam’s expertise comes into play. He helps his students not only in understanding the subject but also advises them on how to overcome the fear of accounts...
Read full article >
Looking for Class 12 Tuition ?
Learn from the Best Tutors on UrbanPro
Are you a Tutor or Training Institute?
Join UrbanPro Today to find students near youThe best tutors for Class 12 Tuition Classes are on UrbanPro
The best Tutors for Class 12 Tuition Classes are on UrbanPro
9
