Solve {ccc}{-14}&{-2}&{-4}{-6}&{-8}&{-12}{-16}&{-10}&{-18}

$$\left| \begin{array} { c c c } { - 14 } & { - 2 } & { - 4 } \\ { - 6 } & { - 8 } & { - 12 } \\ { - 16 } & { - 10 } & { - 18 } \end{array} \right|$$

Evaluate

$$-232$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}-14&-2&-4\\-6&-8&-12\\-16&-10&-18\end{matrix}\right))$$

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

$$\left(\begin{matrix}-14&-2&-4&-14&-2\\-6&-8&-12&-6&-8\\-16&-10&-18&-16&-10\end{matrix}\right)$$

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

$$-14\left(-8\right)\left(-18\right)-2\left(-12\right)\left(-16\right)-4\left(-6\right)\left(-10\right)=-2640$$

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

$$-16\left(-8\right)\left(-4\right)-10\left(-12\right)\left(-14\right)-18\left(-6\right)\left(-2\right)=-2408$$

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

$$-2640-\left(-2408\right)$$

Subtract -2408 from -2640.

$$-232$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}-14&-2&-4\\-6&-8&-12\\-16&-10&-18\end{matrix}\right))$$

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.

$$-14det(\left(\begin{matrix}-8&-12\\-10&-18\end{matrix}\right))-\left(-2det(\left(\begin{matrix}-6&-12\\-16&-18\end{matrix}\right))\right)-4det(\left(\begin{matrix}-6&-8\\-16&-10\end{matrix}\right))$$

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

$$-14\left(-8\left(-18\right)-\left(-10\left(-12\right)\right)\right)-\left(-2\left(-6\left(-18\right)-\left(-16\left(-12\right)\right)\right)\right)-4\left(-6\left(-10\right)-\left(-16\left(-8\right)\right)\right)$$

Simplify.

$$-14\times 24-\left(-2\left(-84\right)\right)-4\left(-68\right)$$

Add the terms to obtain the final result.

$$-232$$

Factor

$$-232$$