JEE Matrices (1)

Algebra Level 4

A \text A and B \text B are two square matrices such that A 2 B = BA \text A^2 \text B = \text{BA} .

If value of (AB) 10 \text{(AB)}^{10} can be expressed in the form A k B 10 \text A^k \text B^{10} , find the value of k + 1 128 \dfrac {k+1}{128} .

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Prakhar Bindal by Prakhar Bindal , 22, India

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2 solutions

Aniket Sanghi
Feb 19, 2017

By Induction you will find that

( A B ) n = A a n B n (AB)^n = {A}^{a_n} B^n ( where , n is a natural number)

Where a n = 2 a n − 1 + 1 a_n = 2 {a}_{n - 1} + 1

9 Helpful 0 Interesting 0 Brilliant 0 Confused
Samarth Agarwal
Feb 20, 2017

( A B ) n = A 2 n − 1 B n (AB)^{n}=A^{2^n-1}B^n .... Cool result and good question!! (+1)

4 Helpful 0 Interesting 0 Brilliant 0 Confused

typo B^n rest perfect! . yeah thanks :)

Prakhar Bindal - 4 years, 3 months ago

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Done thanks!!

Samarth Agarwal - 4 years, 3 months ago

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Hey bro check ur mail

Samarth Agarwal - 4 years, 3 months ago

done replied!

Prakhar Bindal - 4 years, 3 months ago

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