Question

If A, Band C are the angles of a triangle, then the determinant
$\left|\begin{array}{ccc}-1& \cos C& \cos B\\ \cos C& -1& \cos A\\ \cos B& \cos A& -1\end{array}\right|$
is equal to

• A. 0
• B. -1
• C. 1
• D. none of these

Answer 1

Answer: (A)

0

$\left|\begin{array}{ccc}-1& \cos C& \cos B\\ \cos C& -1& \cos A\\ \cos B& \cos A& -1\end{array}\right|$

$=-1(1-{\cos }^{2}A)+\cos C(\cos C+\cos A\cos B)+\cos B(\cos A\cos C-\cos B)$

$=-1+{\cos }^{2}A+{\cos }^{2}B+{\cos }^{2}C+2\cos A\cos B\cos C⟶1$

Given A, B,C are angles of a triangle

Condition $A=B=C=60°$ is possible

$\cos 60=\frac{1}{2}$

$=-1+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+2\left(\frac{1}{2}\right)\left(\frac{1}{2}\right)\left(\frac{1}{2}\right)$

$=0$

$det=0$

Answer option A