Solve {ccc}{2}&{-3}&{-1}{4}&{-6}&{3}{-9}&{5}&{-4}

$$\left| \begin{array} { c c c } { 2 } & { - 3 } & { - 1 } \\ { 4 } & { - 6 } & { 3 } \\ { - 9 } & { 5 } & { - 4 } \end{array} \right|$$

Evaluate

$$85$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}2&-3&-1\\4&-6&3\\-9&5&-4\end{matrix}\right))$$

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

$$\left(\begin{matrix}2&-3&-1&2&-3\\4&-6&3&4&-6\\-9&5&-4&-9&5\end{matrix}\right)$$

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

$$2\left(-6\right)\left(-4\right)-3\times 3\left(-9\right)-4\times 5=109$$

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

$$-9\left(-6\right)\left(-1\right)+5\times 3\times 2-4\times 4\left(-3\right)=24$$

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

$$109-24$$

Subtract 24 from 109.

$$85$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}2&-3&-1\\4&-6&3\\-9&5&-4\end{matrix}\right))$$

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.

$$2det(\left(\begin{matrix}-6&3\\5&-4\end{matrix}\right))-\left(-3det(\left(\begin{matrix}4&3\\-9&-4\end{matrix}\right))\right)-det(\left(\begin{matrix}4&-6\\-9&5\end{matrix}\right))$$

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

$$2\left(-6\left(-4\right)-5\times 3\right)-\left(-3\left(4\left(-4\right)-\left(-9\times 3\right)\right)\right)-\left(4\times 5-\left(-9\left(-6\right)\right)\right)$$

Simplify.

$$2\times 9-\left(-3\times 11\right)-\left(-34\right)$$

Add the terms to obtain the final result.

$$85$$

Factor

$$5\times 17$$