Solve {ccc}{-1}&{2}&{4}{-2}&{3}&{9}{-5}&{-2}&{2}

$$\left| \begin{array} { c c c } { - 1 } & { 2 } & { 4 } \\ { - 2 } & { 3 } & { 9 } \\ { - 5 } & { - 2 } & { 2 } \end{array} \right|$$

Evaluate

$$-30$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}-1&2&4\\-2&3&9\\-5&-2&2\end{matrix}\right))$$

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

$$\left(\begin{matrix}-1&2&4&-1&2\\-2&3&9&-2&3\\-5&-2&2&-5&-2\end{matrix}\right)$$

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

$$-3\times 2+2\times 9\left(-5\right)+4\left(-2\right)\left(-2\right)=-80$$

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

$$-5\times 3\times 4-2\times 9\left(-1\right)+2\left(-2\right)\times 2=-50$$

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

$$-80-\left(-50\right)$$

Subtract -50 from -80.

$$-30$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}-1&2&4\\-2&3&9\\-5&-2&2\end{matrix}\right))$$

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.

$$-det(\left(\begin{matrix}3&9\\-2&2\end{matrix}\right))-2det(\left(\begin{matrix}-2&9\\-5&2\end{matrix}\right))+4det(\left(\begin{matrix}-2&3\\-5&-2\end{matrix}\right))$$

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

$$-\left(3\times 2-\left(-2\times 9\right)\right)-2\left(-2\times 2-\left(-5\times 9\right)\right)+4\left(-2\left(-2\right)-\left(-5\times 3\right)\right)$$

Simplify.

$$-24-2\times 41+4\times 19$$

Add the terms to obtain the final result.

$$-30$$

Factor

$$-30$$