Question

Evaluate the following determinants:

(i) $\left|\begin{array}{cc}x& -7\\ x& 5x+1\end{array}\right|$

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

(iii) $\left|\begin{array}{cc}\cos 15°& \sin 15°\\ \sin 75°& \cos 75°\end{array}\right|$

(iv) $\left|\begin{array}{cc}a+ib& c+id\\ -c+id& a-ib\end{array}\right|$

Answer 1

(i)
$$\begin{gathered} ∆=x(5x+1)+7x=5x^2+x+7x \\ =5x^2+8x \end{gathered}$$

(ii)
$$\begin{gathered} ∆={\cos }^2\theta -\left(-{\sin }^2\theta \right) \\ ={\cos }^2\theta +{\sin }^2\theta =1 \end{gathered}$$

(iii)
$$\begin{gathered} ∆=\cos 15°\cos 75°-\sin 15°\sin 75° \\ =\cos 15°\cos 75°-\sin (90°-75°)\sin (90°-15°)\left[\because \sin \left(90°-\theta \right)=\cos \theta \right] \\ =\cos 15°\cos 75°-\cos 75°\cos 15° \\ =\cos 15°\cos 75°-\cos 15°\cos 75°=0 \end{gathered}$$

(iv)
$$\begin{gathered} ∆=a^2-i^2{b}^2-\left(i^2{d}^2-c^2\right) \\ =a^2-i^2{b}^2-i^2{d}^2+c^2 \\ =a^2+c^2-i^2\left(b^2+d^2\right)\left[\because i^2=-1\right] \\ =a^2+c^2+b^2+d^2 \end{gathered}$$