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

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

Evaluate

$$-65$$

Steps Using Sarrus' Rule

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

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

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

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

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

$$-3\times 2\left(-5\right)-5\left(-4\right)+2\times 3\times 6=86$$

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

$$2\times 2+6\left(-4\right)\left(-3\right)-5\times 3\left(-5\right)=151$$

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

$$86-151$$

Subtract 151 from 86.

$$-65$$

Steps Using Expansion by Minors

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

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

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.

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

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

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

Simplify.

$$-3\times 14-\left(-5\left(-11\right)\right)+2\times 16$$

Add the terms to obtain the final result.

$$-65$$

Factor

$$-65$$