Solve {ccc}{-2}&{3}&{-5}{-5}&{6}&{-8}{22}&{2}&{25}

$$\left| \begin{array} { c c c } { - 2 } & { 3 } & { - 5 } \\ { - 5 } & { 6 } & { - 8 } \\ { 22 } & { 2 } & { 25 } \end{array} \right|$$

Evaluate

$$225$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}-2&3&-5\\-5&6&-8\\22&2&25\end{matrix}\right))$$

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

$$\left(\begin{matrix}-2&3&-5&-2&3\\-5&6&-8&-5&6\\22&2&25&22&2\end{matrix}\right)$$

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

$$-2\times 6\times 25+3\left(-8\right)\times 22-5\left(-5\right)\times 2=-778$$

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

$$22\times 6\left(-5\right)+2\left(-8\right)\left(-2\right)+25\left(-5\right)\times 3=-1003$$

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

$$-778-\left(-1003\right)$$

Subtract -1003 from -778.

$$225$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}-2&3&-5\\-5&6&-8\\22&2&25\end{matrix}\right))$$

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.

$$-2det(\left(\begin{matrix}6&-8\\2&25\end{matrix}\right))-3det(\left(\begin{matrix}-5&-8\\22&25\end{matrix}\right))-5det(\left(\begin{matrix}-5&6\\22&2\end{matrix}\right))$$

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

$$-2\left(6\times 25-2\left(-8\right)\right)-3\left(-5\times 25-22\left(-8\right)\right)-5\left(-5\times 2-22\times 6\right)$$

Simplify.

$$-2\times 166-3\times 51-5\left(-142\right)$$

Add the terms to obtain the final result.

$$225$$

Factor

$$3^{2}\times 5^{2}$$