QJEEAdv15Maaef1: JEE Advanced 2015 Mathematics - Class 12 JEEAdv15Maaef1 Solution: Let X and Y be two arbitrary, 3 3, non-zero, skew-symmetric…

Given,

${X^T} = - X,{Y^T} = - Y,{Z^T} = Z$

(a) Let $P = {Y^3}{Z^4} - {Z^4}{Y^3}$

Then, ${P^T} = {({Y^3}{Z^4})^T} - {({Z^4}{Y^3})^T}$

$= {({Z^T})^4}{({Y^T})^3} - {({Y^T})^3}{({Z^T})^4}$

$= - {Z^4}{Y^3} + {Y^3}{Z^4} = P$

$\therefore$ P is symmetric matrix.

(b) Let $P = {X^{44}} + {Y^{44}}$

Then, ${P^T} = {({X^T})^{44}} + {({Y^T})^{44}}$

$= {X^{44}} + {Y^{44}} = P$

$\therefore$ P is symmetric matrix.

(c) Let $P = {X^4}{Z^3} - {Z^3}{X^4}$

Then, ${P^T} = {({X^4}{Z^3})^T} - {({Z^3}{X^4})^T}$

$= {({Z^T})^3}{({X^T})^4} - {({X^T})^4}{({Z^T})^3}$

$= {Z^3}{X^4} - {X^4}{Z^3}$

$= - P$

$\therefore$ P is skew-symmetric matrix.

(d) Let $P = {X^{23}} + {Y^{23}}$

Then, ${P^T} = {({X^T})^{23}} + {({Y^T})^{23}}$

$= - {X^{23}} - {Y^{23}}$

$= - P$

$\therefore$ P is skew-symmetric matrix.