JEE Advanced 2013 · Paper 1 · Mathematics Q52

MathematicsMultiple correct+4 / -1Question 52 of 60

For $3\times3$ matrices $M$ and $N$ , which of the following statement(s) is (are) NOT correct?

N^TMN

is symmetric or skew symmetric, according as

M

is symmetric or skew symmetric

MN-NM

is skew symmetric for all symmetric matrices

M

and

N

MN

is symmetric for all symmetric matrices

M

and

N

(\operatorname{adj}M)(\operatorname{adj}N)=\operatorname{adj}(MN)

for all invertible matrices

M

and

N

Answer and worked solution

Official answer: C, D

(A) $(N^TMN)^T=N^TM^TN=\pm N^TMN$ as $M^T=\pm M$ . Correct.
(B) $(MN-NM)^T=NM-MN$ . Correct.
(C) $(MN)^T=NM$ , equal to $MN$ only if they commute. Not correct.
(D) $\operatorname{adj}(MN)=(\operatorname{adj}N)(\operatorname{adj}M)$ , in reverse order. Not correct in general.

(C), (D)

Topics:Special matrices Adjoint and inverse

JEE Advanced 2013 · Paper 1, Mathematics question 52