JEE Advanced Matrices and Determinants 1

Algebra Level 3

Suppose A A and B B are two non singular matrices such that A B = B A 2 AB=BA^2 and B 5 = I B^5=I , then

This is a part of My Picks for JEE Advanced 2
A 32 = I A^{32}=I A 31 = I A^{31}=I A 30 = I A^{30}=I A 50 = I A^{50}=I

Pranjal Jain by Pranjal Jain , 23, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Aakarshit Uppal
Apr 21, 2015

It is given that A B = B A 2 AB = BA^2

Post-multiplying by B on both side,

A B 2 = B A 2 B = B A ( A B ) = B A ( B A 2 ) = B ( A B ) A 2 = B ( B A 2 ) A 2 = B 2 A 4 AB^2 = BA^2B = BA(AB) = BA(BA^2) = B(AB)A^2 = B(BA^2)A^2 = B^2A^4

(By using the given property) ( Parentheses have been used to highlight the substitutions)

Again Post-multiplying by B on both sides,

A B 3 = B 2 A 4 B = B 2 A 3 ( A B ) = B 2 A 3 ( B A 2 ) = B 2 A 2 ( A B ) A 2 = B 2 A 2 ( B A 2 ) A 2 = B 2 A ( A B ) A 4 = B 2 A ( B A 2 ) A 4 = B 2 ( A B ) A 6 = B 2 ( B A 2 ) A 6 = B 3 A 8 AB^3 = B^2A^4B = B^2A^3(AB) = B^2A^3(BA^2) = B^2A^2(AB)A^2 = B^2A^2(BA^2)A^2 = B^2A(AB)A^4 = B^2A(BA^2)A^4 = B^2(AB)A^6 = B^2(BA^2)A^6 = B^3A^8

Thus, upon n-1 post multiplications of B on both sides, we have

A B n = B n A 2 n AB^n = B^nA^{2^n}

Putting n=5, we get

A B 5 = B 5 A 32 AB^5 = B^5A^{32} ⇒ A = A 32 \Rightarrow A = A^{32} (As B 5 = I B^5 = I )

Post-multiplying by A − 1 A^{-1} on both sides (which exists as A A is non-singular)

A A − 1 = A 32 A − 1 = A 31 ( A A − 1 ) AA^{-1} = A^{32}A^{-1} = A^{31}(AA^{-1}) ⇒ I = A 31 I \Rightarrow I = A^{31}I ⇒ A 31 = I \Rightarrow \color{#20A900}{\boxed{ A^{31} = I }}

37 Helpful 6 Interesting 10 Brilliant 6 Confused

Words would fall short if I were to describe this solution using them.Fabulous!How did this method occur to you.I would be delighted to learn that from you as I am a rookie in Matrices and determinants.Thank you.

pranav jangir - 5 years, 7 months ago

Log in to reply

Nice solution

Ram Sita - 3 years, 2 months ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...