Solve {lll}{19}&{6}&{10}{4}&{6}&{-10}{10}&{-5}&{20}

$$\left| \begin{array} { l l l } { 19 } & { 6 } & { 10 } \\ { 4 } & { 6 } & { - 10 } \\ { 10 } & { - 5 } & { 20 } \end{array} \right|$$

Evaluate

$$-550$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}19&6&10\\4&6&-10\\10&-5&20\end{matrix}\right))$$

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

$$\left(\begin{matrix}19&6&10&19&6\\4&6&-10&4&6\\10&-5&20&10&-5\end{matrix}\right)$$

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

$$19\times 6\times 20+6\left(-10\right)\times 10+10\times 4\left(-5\right)=1480$$

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

$$10\times 6\times 10-5\left(-10\right)\times 19+20\times 4\times 6=2030$$

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

$$1480-2030$$

Subtract 2030 from 1480.

$$-550$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}19&6&10\\4&6&-10\\10&-5&20\end{matrix}\right))$$

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.

$$19det(\left(\begin{matrix}6&-10\\-5&20\end{matrix}\right))-6det(\left(\begin{matrix}4&-10\\10&20\end{matrix}\right))+10det(\left(\begin{matrix}4&6\\10&-5\end{matrix}\right))$$

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

$$19\left(6\times 20-\left(-5\left(-10\right)\right)\right)-6\left(4\times 20-10\left(-10\right)\right)+10\left(4\left(-5\right)-10\times 6\right)$$

Simplify.

$$19\times 70-6\times 180+10\left(-80\right)$$

Add the terms to obtain the final result.

$$-550$$

Factor

$$-550$$