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

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

Evaluate

$$-8$$

Steps Using Sarrus' Rule

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

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

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

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

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

$$3\times 3\left(-3\right)-\left(-4\right)+2\left(-5\right)\times 3=-53$$

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

$$3\times 2+3\left(-4\right)\times 3-3\left(-5\right)\left(-1\right)=-45$$

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

$$-53-\left(-45\right)$$

Subtract -45 from -53.

$$-8$$

Steps Using Expansion by Minors

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

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

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.

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

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

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

Simplify.

$$3\times 3-\left(-19\right)+2\left(-18\right)$$

Add the terms to obtain the final result.

$$-8$$

Factor

$$-8$$