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.