Solve {ccc}{15}&{29}&{14}{16}&{19}&{17}{33}&{39}&{38}

$$\left| \begin{array} { c c c } { 15 } & { 29 } & { 14 } \\ { 16 } & { 19 } & { 17 } \\ { 33 } & { 39 } & { 38 } \end{array} \right|$$

Evaluate

$$-520$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}15&29&14\\16&19&17\\33&39&38\end{matrix}\right))$$

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

$$\left(\begin{matrix}15&29&14&15&29\\16&19&17&16&19\\33&39&38&33&39\end{matrix}\right)$$

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

$$15\times 19\times 38+29\times 17\times 33+14\times 16\times 39=35835$$

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

$$33\times 19\times 14+39\times 17\times 15+38\times 16\times 29=36355$$

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

$$35835-36355$$

Subtract 36355 from 35835.

$$-520$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}15&29&14\\16&19&17\\33&39&38\end{matrix}\right))$$

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.

$$15det(\left(\begin{matrix}19&17\\39&38\end{matrix}\right))-29det(\left(\begin{matrix}16&17\\33&38\end{matrix}\right))+14det(\left(\begin{matrix}16&19\\33&39\end{matrix}\right))$$

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

$$15\left(19\times 38-39\times 17\right)-29\left(16\times 38-33\times 17\right)+14\left(16\times 39-33\times 19\right)$$

Simplify.

$$15\times 59-29\times 47+14\left(-3\right)$$

Add the terms to obtain the final result.

$$-520$$

Factor

$$-520$$