JEE Advanced 2015 · Paper 1 · Mathematics Q49

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MathematicsMultiple correct+4 / -2Question 49 of 60

Let $X$ and $Y$ be two arbitrary, $3\times3$ , non-zero, skew-symmetric matrices and $Z$ be an arbitrary $3\times3$ , non-zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric?

Y^3Z^4-Z^4Y^3

X^{44}+Y^{44}

X^4Z^3-Z^3X^4

X^{23}+Y^{23}

Answer and worked solution

Official answer: C, D

Transposes. $X^T=-X$ , $Y^T=-Y$ and $Z^T=Z$ .

(A) $(Y^3Z^4-Z^4Y^3)^T=-Z^4Y^3+Y^3Z^4$ , the same matrix, so it is symmetric. False.
(B) Even powers of skew matrices are symmetric. False.
(C) $(X^4Z^3-Z^3X^4)^T=Z^3X^4-X^4Z^3$ , the negative, so it is skew. True.
(D) Odd powers of skew matrices are skew, and so is their sum. True.

(C), (D)

Topics:Special matrices

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JEE Advanced 2015 · Paper 1, Mathematics question 49