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Determinants and Matrices — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsDeterminants and Matrices1 Mark
Question
Find the values of the determinants $:\left|\begin{array}{lll}\mathrm{a} & \mathrm{h} & \mathrm{g} \\ \mathrm{h} & \mathrm{b} & \mathrm{f} \\ \mathrm{g} & \mathrm{f} & \mathrm{c}\end{array}\right|$
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Answer

$\left|\begin{array}{lll}a & h & g \\ h & b & f \\ g & f & c\end{array}\right|=a\left|\begin{array}{ll}b & f \\ f & c\end{array}\right|-h\left|\begin{array}{cc}h & f \\ g & c\end{array}\right|+g\left|\begin{array}{cc}h & b \\ g & f\end{array}\right|$
$ =a\left(b c-f^2\right)-h(h c-g f)+g(h f-g b)$
$ =a b c-a f^2-h^2 c+f g h+f g h-g^2 b$
$ =a b c+2 f g h-a f^2-b g^2-c h^2$
$ =(-15)-(-4)(7)$
$ =-30+28$
$ =-2$

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