(i) $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{\cos \theta }&{ - \sin \theta }\\{\sin \theta }&{\cos \theta }\end{array}} \right|$ (ii) $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{{x^2} - x + 1}&{x - 1}\\{x + 1}&{x + 1}\end{array}} \right|$

(i)$\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{\cos \theta }&{ - \sin \theta }\\{\sin \theta }&{\cos \theta }\end{array}} \right|$

$= \cos \theta \times \cos \theta - (\sin \theta ) \times ( - \sin \theta ) = {\cos ^2}\theta + {\sin ^2}\theta = 1$

(ii) $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{{x^2} - x + 1}&{x - 1}\\{x + 1}&{x + 1}\end{array}} \right| = ({x^2} - x + 1)(x + 1)(x + 1)(x - 1)$

$= {x^3} + 1 - ({x^2} - 1) = {x^3} + 1 - {x^2} + 1 = {x^3} - {x^2} + 2$