Let α, β, γ be three real numbers and A=1cos⁡(β−α)cos⁡(γ−α)cos⁡(α−β)1cos⁡(γ−β)cos⁡(α−γ)cos⁡(β−γ)1 then

# Let  be three real numbers and $A=\left[\begin{array}{ccc}1& \mathrm{cos}\left(\beta -\alpha \right)& \mathrm{cos}\left(\gamma -\alpha \right)\\ \mathrm{cos}\left(\alpha -\beta \right)& 1& \mathrm{cos}\left(\gamma -\beta \right)\\ \mathrm{cos}\left(\alpha -\gamma \right)& \mathrm{cos}\left(\beta -\gamma \right)& 1\end{array}\right]$ then

1. A

$A$is singular

2. B

$A$is non-singular

3. C

$A$is orthogonal

4. D

none of these

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### Solution:

$|A|=\left|\begin{array}{ccc}1& \mathrm{cos}\left(\beta -\alpha \right)& \mathrm{cos}\left(\gamma -\alpha \right)\\ \mathrm{cos}\left(\alpha -\beta \right)& 1& \mathrm{cos}\left(\gamma -\beta \right)\\ \mathrm{cos}\left(\alpha -\gamma \right)& \mathrm{cos}\left(\beta -\gamma \right)& 1\end{array}\right|$
$=\left|\begin{array}{ccc}\mathrm{cos}\alpha & \mathrm{sin}\alpha & 0\\ \mathrm{cos}\beta & \mathrm{sin}\beta & 0\\ \mathrm{cos}\gamma & \mathrm{cos}\gamma & 0\end{array}\right|\left|\begin{array}{ccc}\mathrm{cos}\alpha & \mathrm{sin}\alpha & 0\\ \mathrm{cos}\beta & \mathrm{sin}\beta & 0\\ \mathrm{cos}\gamma & \mathrm{sin}\gamma & 0\end{array}\right|=0$
is singular.