Solve {ccc}{-2}&{3}&{4}{8}&{-12}&{20}{12}&{18}&{-30}

$$\left| \begin{array} { c c c } { - 2 } & { 3 } & { 4 } \\ { 8 } & { - 12 } & { 20 } \\ { 12 } & { 18 } & { - 30 } \end{array} \right|$$

Evaluate

$$2592$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}-2&3&4\\8&-12&20\\12&18&-30\end{matrix}\right))$$

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

$$\left(\begin{matrix}-2&3&4&-2&3\\8&-12&20&8&-12\\12&18&-30&12&18\end{matrix}\right)$$

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

$$-2\left(-12\right)\left(-30\right)+3\times 20\times 12+4\times 8\times 18=576$$

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

$$12\left(-12\right)\times 4+18\times 20\left(-2\right)-30\times 8\times 3=-2016$$

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

$$576-\left(-2016\right)$$

Subtract -2016 from 576.

$$2592$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}-2&3&4\\8&-12&20\\12&18&-30\end{matrix}\right))$$

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.

$$-2det(\left(\begin{matrix}-12&20\\18&-30\end{matrix}\right))-3det(\left(\begin{matrix}8&20\\12&-30\end{matrix}\right))+4det(\left(\begin{matrix}8&-12\\12&18\end{matrix}\right))$$

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

$$-2\left(-12\left(-30\right)-18\times 20\right)-3\left(8\left(-30\right)-12\times 20\right)+4\left(8\times 18-12\left(-12\right)\right)$$

Simplify.

$$-3\left(-480\right)+4\times 288$$

Add the terms to obtain the final result.

$$2592$$

Factor

$$2^{5}\times 3^{4}$$