Question

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

Answer 1

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

$\left(ii\right)$ $\left|\begin{array}{cc}{x}^2-x+1& x-1\\ x+1& x+1\end{array}\right|$
$=(x^2-x+1)(x+1)-(x+1)(x-1)$
$=(x^2-x+1)(x+1)-(x^2-1)$
$=\left[\right(x^2-x+1)x+(x^2-x+1\left)1\right]-(x^2-1)$
$=(x^3-x^2+x+x^2-x+1)-(x^2-1)$
$=x^3+1-x^2+1$
$=x^3-x^2+2$