Evaluate the determinants in Exercises 1 and 2.

Question:

Evaluate the determinants in Exercises 1 and 2.

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

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

Solution:

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

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

$=\left(x^{2}-x+1\right)(x+1)-(x-1)(x+1)$

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

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

$=x^{3}-x^{2}+2$