Gate 2014

Question

Qus 1

Qus2 

Answer 1

Answer to qno 1 :

A) is false because multiplication of matrices is not commutative..Similarly B) is false bcoz in expansion we come across term QP and PQ which are necessarily not the same..Also determinant property mentioned in option C) is false..And if we consider D) option , then we can see it treats PQ and QP separately..

Hence D) is correct answer

Answer to qno 2 :

We know :

1) Product of eigen values of  matrix  = Det(Matrix)

2) Sum of eigen values of matrix  =  Trace(Matrix)

So let one eigen value  = x

Then other eigen value  =  14 - x [As trace of matrix  = 14]

So determinant  =  product of eigen values  = x(14 - x)   =  14 x - x²

So let y  =  14 x - x² 

Then dy / dx = 0 and d² y / dx² < 0 for maxima condition

So

dy / dx  =  14 - 2 x = 0  == > x = 7

So 

y    =   14 * 7 - 7²  = 49

Hence maximum determinant possible  = 49