Solve {ccc}{-2}&{-4}&{-6}{-8}&{-2}&{0}{2}&{-4}&{6}

$$\left| \begin{array} { c c c } { - 2 } & { - 4 } & { - 6 } \\ { - 8 } & { - 2 } & { 0 } \\ { 2 } & { - 4 } & { 6 } \end{array} \right|$$

Evaluate

$$-384$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}-2&-4&-6\\-8&-2&0\\2&-4&6\end{matrix}\right))$$

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

$$\left(\begin{matrix}-2&-4&-6&-2&-4\\-8&-2&0&-8&-2\\2&-4&6&2&-4\end{matrix}\right)$$

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

$$-2\left(-2\right)\times 6-6\left(-8\right)\left(-4\right)=-168$$

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

$$2\left(-2\right)\left(-6\right)+6\left(-8\right)\left(-4\right)=216$$

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

$$-168-216$$

Subtract 216 from -168.

$$-384$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}-2&-4&-6\\-8&-2&0\\2&-4&6\end{matrix}\right))$$

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.

$$-2det(\left(\begin{matrix}-2&0\\-4&6\end{matrix}\right))-\left(-4det(\left(\begin{matrix}-8&0\\2&6\end{matrix}\right))\right)-6det(\left(\begin{matrix}-8&-2\\2&-4\end{matrix}\right))$$

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

$$-2\left(-2\right)\times 6-\left(-4\left(-8\right)\times 6\right)-6\left(-8\left(-4\right)-2\left(-2\right)\right)$$

Simplify.

$$-2\left(-12\right)-\left(-4\left(-48\right)\right)-6\times 36$$

Add the terms to obtain the final result.

$$-384$$

Factor

$$-384$$