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in Determinants by (42.4k points)
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Find the values of the determinants.

\(\begin{vmatrix}a&h&g\\h&b&f\\g&f&c\end{vmatrix}\)

[(a,h,g) (h,b,f)(g,f,c)]

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\(\begin{vmatrix}a&h&g\\h&b&f\\g&f&c\end{vmatrix}=a\begin{vmatrix}b&f\\f&c\end{vmatrix}-h\begin{vmatrix}h&f\\g&c\end{vmatrix}+g\begin{vmatrix}h&b\\g&f\end{vmatrix}\)

= a(bc – f2 ) – h(hc — gf) + g(hf- gb) 

= abc – af2 – h2 c + fgh + fgh – g2 b

= abc + 2fgh – af2 – bg2 – ch2 

= (-15) – (-4)(7) 

= -30 + 28 

= -2

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