# Can you plz help me in solving this question?

## Multiple options can be correct. Apr 16, 2018

See below.

#### Explanation:

If $Y$ is skew-symmetric then $Y = - {Y}^{\top}$
If $Z$ is symmetric then $Z = {Z}^{\top}$

a)

$- {\left({Y}^{3} {Z}^{4} - {Z}^{4} {Y}^{3}\right)}^{\top} = - \left({Z}^{4} {\left({Y}^{\top}\right)}^{3} - {\left({Y}^{\top}\right)}^{3} {Z}^{4}\right) = {Z}^{4} {Y}^{3} - {Y}^{3} {Z}^{4} = - \left({Y}^{3} {Z}^{4} - {Z}^{4} {Y}^{3}\right)$ hence a) is a symmetric matrix.

b)

${X}^{44} + {Y}^{44} = {\left(- X \cdot {X}^{\top}\right)}^{22} + {\left(- Y \cdot Y\right)}^{22}$ which is a symmetric matrix.

c)

X^4Z^3-Z^3 X^4 = (-X X^top)^2 Z^3-Z^3 (-X X^dot)^2 = (X X^top)^2 Z^3-Z^3 (X X^top)^2 = -((X X^top)^2 Z^3-Z^3 (X X^top)^2)^top hence this is a skew-symmetric matrix

d)

${X}^{23} + {Y}^{23} = {\left(- {X}^{\top}\right)}^{23} + {\left(- {Y}^{\top}\right)}^{23} = - {\left({X}^{\top}\right)}^{23} - {\left({Y}^{\top}\right)}^{23} = - {\left({X}^{23} + {Y}^{23}\right)}^{\top}$ hence d) is skew-symmetric.