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

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

Evaluate

$$110$$

Steps Using Sarrus' Rule

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

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

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

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

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

$$3\times 3\times 3-\left(-\left(-5\right)\right)+4\times 2\times 2=38$$

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

$$-5\times 3\times 4+2\left(-1\right)\times 3+3\times 2\left(-1\right)=-72$$

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

$$38-\left(-72\right)$$

Subtract -72 from 38.

$$110$$

Steps Using Expansion by Minors

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

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

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.

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

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

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

Simplify.

$$3\times 11-\left(-1\right)+4\times 19$$

Add the terms to obtain the final result.

$$110$$

Factor

$$2\times 5\times 11$$