Solve {rrr}{8}&{-2}&{-10}{2}&{-3}&{-2}{6}&{3}&{8}

$$\left| \begin{array} { r r r } { 8 } & { - 2 } & { - 10 } \\ { 2 } & { - 3 } & { - 2 } \\ { 6 } & { 3 } & { 8 } \end{array} \right|$$

Evaluate

$$-328$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}8&-2&-10\\2&-3&-2\\6&3&8\end{matrix}\right))$$

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

$$\left(\begin{matrix}8&-2&-10&8&-2\\2&-3&-2&2&-3\\6&3&8&6&3\end{matrix}\right)$$

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

$$8\left(-3\right)\times 8-2\left(-2\right)\times 6-10\times 2\times 3=-228$$

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

$$6\left(-3\right)\left(-10\right)+3\left(-2\right)\times 8+8\times 2\left(-2\right)=100$$

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

$$-228-100$$

Subtract 100 from -228.

$$-328$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}8&-2&-10\\2&-3&-2\\6&3&8\end{matrix}\right))$$

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.

$$8det(\left(\begin{matrix}-3&-2\\3&8\end{matrix}\right))-\left(-2det(\left(\begin{matrix}2&-2\\6&8\end{matrix}\right))\right)-10det(\left(\begin{matrix}2&-3\\6&3\end{matrix}\right))$$

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

$$8\left(-3\times 8-3\left(-2\right)\right)-\left(-2\left(2\times 8-6\left(-2\right)\right)\right)-10\left(2\times 3-6\left(-3\right)\right)$$

Simplify.

$$8\left(-18\right)-\left(-2\times 28\right)-10\times 24$$

Add the terms to obtain the final result.

$$-328$$

Factor

$$-328$$