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

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

Evaluate

$$-29$$

Steps Using Sarrus' Rule

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

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

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

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

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

$$-2\times 4\left(-2\right)-\left(-2\times 4\right)+5\times 3\times 3=69$$

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

$$4\times 4\times 5+3\left(-2\right)\left(-2\right)-2\times 3\left(-1\right)=98$$

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

$$69-98$$

Subtract 98 from 69.

$$-29$$

Steps Using Expansion by Minors

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

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

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.

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

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

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

Simplify.

$$-2\left(-2\right)-\left(-2\right)+5\left(-7\right)$$

Add the terms to obtain the final result.

$$-29$$

Factor

$$-29$$