JEE Advanced Matrices and Determinants 2

Algebra Level 4

Find the sum of all values of n n such that atleast one pair of two matrices ( A , B ) (A,B) of order n n satisfies A B − B A = I AB-BA=I .

Note that A A and B B are matrices and not scalars.

Enter − 1 -1 if no n n exists.

This is a part of My Picks for JEE Advanced 2


The answer is -1.

Pranjal Jain by Pranjal Jain , 23, India

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1 solution

Arpan Banerjee
Feb 25, 2015

Taking trace of the matrices on both sides yields 0 on L.H.S. (since trace(AB)=trace(BA) ) and 'n' on the R.H.S.

The trace of AB is 

           A(1,1)B(1,1)+A(1,2)B(2,1)+...+A(1,n)B(n,1),
           A(2,1)B(1,2)+A(2,2)B(2,2)+...+A(2,n)B(n,2),
           ..........................................
           A(n,1)B(1,n)+A(n,2)B(2,n)+...+A(n,n)B(n,n)

Trace of BA is 

           A(1,1)B(1,1)+A(2,1)B(1,2)+...+A(n,1)B(1,n)+
           A(1,2)B(2,1)+A(2,2)B(2,2)+...+A(n,2)B(2,n)+
           ...........................................
           A(1,n)B(n,1)+A(2,n)B(n,2)+...+A(n,n)B(n,n)

This is infact the same sum, only rows and columns have been interchanged. So, trace(AB) = trace(BA)

Since traces differ on both sides, no solution exists.

8 Helpful 0 Interesting 0 Brilliant 0 Confused

Can you add proof of the fact that trace of A B AB is equal to that of B A BA ?

Perfect now, @Arpan Banerjee

Pranjal Jain - 6 years, 3 months ago

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