JEE Main · 2012 · 29 Apr 2012 · Offline · Matrices

Let

P

and

Q

3 \times 3

matrices with

P \neq = Q

. If

P^{3} = Q^{3}

and

P^{2} Q = Q^{2} P

, then determinant of (

P^{2} + Q^{2}

) is equal to

Answer: (C)

Solution

From

P^{3} = Q^{3}

and

P^{2} Q = Q^{2} P

, subtract the second pair of products from the first to get

P^{3} - P^{2} Q = Q^{3} - Q^{2} P

, i.e.

P^{2} (P - Q) = - Q^{2} (P - Q)

. Bringing everything to one side,

(P^{2} + Q^{2}) (P - Q) = O

. Since

P \neq = Q

the factor

P - Q

is not the zero matrix, so the product can vanish only if

P^{2} + Q^{2}

is singular. Therefore

det (P^{2} + Q^{2}) = 0

JEE Main · 2012 · 29 Apr 2012 · Offline · Single correct option