Question

Evaluate the following deteminants :
i)$\left|\begin{array}{cc}x& -7\\ x& 5x+1\end{array}\right|$
ii)$\left|\begin{array}{cc}\cos {15}^{\circ }& \sin {15}^{\circ }\\ \sin {75}^{\circ }& \cos {75}^{\circ }\end{array}\right|$

Answer 1

i)$\left|\begin{array}{cc}x& -7\\ x& 5x+1\end{array}\right|=5x^2+x+7x=5x^2+8x=x(5x+8)$

ii)$\left|\begin{array}{cc}\cos {15}^o& \sin {15}^o\\ \sin {75}^o& \cos {75}^o\end{array}\right|$

$=\left|\begin{array}{cc}\cos {15}^o& \sin {15}^o\\ \cos {15}^o& \sin {15}^o\end{array}\right|=0$ [Since $\sin {15}^o=\cos {75}^o$ and $\cos {15}^o=\sin {75}^o]$