Question

Evaluate the determinants.

$\left|\begin{array}{cc}cos\theta & -sin\theta \\ sin\theta & cos\theta \end{array}\right|$

$\left|\begin{array}{cc}{x}^2-x{+}_1& x-1\\ x+1& x+1\end{array}\right|$

Answer 1

$\left|\begin{array}{cc}cos\theta & -sin\theta \\ sin\theta & cos\theta \end{array}\right|=\left(cos\theta \right)\left(cos\theta \right)-\left(sin\theta \right)(-sin\theta )$
$=co{s}^2\theta +si{n}^2\theta =1(\because si{n}^2\theta +co{s}^2\theta =1)$

$\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^3-x^2+x+x^2-x+1-(x^2-1)=x^3+1(x^2-1)=x^3-x^2+2$