Solve {lll}{a}&{h}&{g}{h}&{b}&{f}{g}&{g}&{c}

$$\left| \begin{array} { l l l } { a } & { h } & { g } \\ { h } & { b } & { f } \\ { g } & { g } & { c } \end{array} \right|$$

Evaluate

$$abc-afg-bg^{2}-ch^{2}+fgh+hg^{2}$$

Steps Using Sarrus' Rule

Find the determinant of the matrix using the method of diagonals.

$$det(\left(\begin{matrix}a&h&g\\h&b&f\\g&g&c\end{matrix}\right))$$

Extend the original matrix by repeating the first two columns as the fourth and fifth columns.

$$\left(\begin{matrix}a&h&g&a&h\\h&b&f&h&b\\g&g&c&g&g\end{matrix}\right)$$

Starting at the upper left entry, multiply down along the diagonals, and add the resulting products.

$$abc+hfg+ghg=abc+fgh+hg^{2}$$

Starting at the lower left entry, multiply up along the diagonals, and add the resulting products.

$$gbg+gfa+chh=afg+bg^{2}+ch^{2}$$

Subtract the sum of the upward diagonal products from the sum of the downward diagonal products.

$$abc+fgh+hg^{2}-\left(afg+bg^{2}+ch^{2}\right)$$

Subtract bg^{2}+gfa+ch^{2} from abc+hfg+hg^{2}.

$$abc-afg-bg^{2}-ch^{2}+fgh+hg^{2}$$

Steps Using Expansion by Minors

Find the determinant of the matrix using the method of expansion by minors (also known as expansion by cofactors).

$$det(\left(\begin{matrix}a&h&g\\h&b&f\\g&g&c\end{matrix}\right))$$

To expand by minors, multiply each element of the first row by its minor, which is the determinant of the 2\times 2 matrix created by deleting the row and column containing that element, then multiply by the element's position sign.

$$adet(\left(\begin{matrix}b&f\\g&c\end{matrix}\right))-hdet(\left(\begin{matrix}h&f\\g&c\end{matrix}\right))+gdet(\left(\begin{matrix}h&b\\g&g\end{matrix}\right))$$

For the 2\times 2 matrix \left(\begin{matrix}a&b\\c&d\end{matrix}\right), the determinant is ad-bc.

$$a\left(bc-gf\right)-h\left(hc-gf\right)+g\left(hg-gb\right)$$

Simplify.

$$a\left(bc-fg\right)-h\left(ch-fg\right)+gg\left(h-b\right)$$

Add the terms to obtain the final result.

$$a\left(bc-fg\right)-h\left(ch-fg\right)+\left(h-b\right)g^{2}$$

Integrate w.r.t. a

$$\frac{bca^{2}}{2}-\frac{fga^{2}}{2}+afgh+ahg^{2}-abg^{2}-ach^{2}+С$$