Solve {ccc}{18}&{-1}&{-1}{10}&{3}&{-2}{-22}&{-2}&{3}

$$\left| \begin{array} { c c c } { 18 } & { - 1 } & { - 1 } \\ { 10 } & { 3 } & { - 2 } \\ { - 22 } & { - 2 } & { 3 } \end{array} \right|$$

Evaluate

$$30$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}18&-1&-1\\10&3&-2\\-22&-2&3\end{matrix}\right))$$

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

$$\left(\begin{matrix}18&-1&-1&18&-1\\10&3&-2&10&3\\-22&-2&3&-22&-2\end{matrix}\right)$$

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

$$18\times 3\times 3-\left(-2\left(-22\right)\right)-10\left(-2\right)=138$$

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

$$-22\times 3\left(-1\right)-2\left(-2\right)\times 18+3\times 10\left(-1\right)=108$$

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

$$138-108$$

Subtract 108 from 138.

$$30$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}18&-1&-1\\10&3&-2\\-22&-2&3\end{matrix}\right))$$

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.

$$18det(\left(\begin{matrix}3&-2\\-2&3\end{matrix}\right))-\left(-det(\left(\begin{matrix}10&-2\\-22&3\end{matrix}\right))\right)-det(\left(\begin{matrix}10&3\\-22&-2\end{matrix}\right))$$

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

$$18\left(3\times 3-\left(-2\left(-2\right)\right)\right)-\left(-\left(10\times 3-\left(-22\left(-2\right)\right)\right)\right)-\left(10\left(-2\right)-\left(-22\times 3\right)\right)$$

Simplify.

$$18\times 5-\left(-\left(-14\right)\right)-46$$

Add the terms to obtain the final result.

$$30$$

Factor

$$2\times 3\times 5$$