Solve {ccc}{-1}&{-2}&{5}{1}&{3}&{6}{-7}&{3}&{2}

$$\left| \begin{array} { c c c } { - 1 } & { - 2 } & { 5 } \\ { 1 } & { 3 } & { 6 } \\ { - 7 } & { 3 } & { 2 } \end{array} \right|$$

Evaluate

$$220$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}-1&-2&5\\1&3&6\\-7&3&2\end{matrix}\right))$$

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

$$\left(\begin{matrix}-1&-2&5&-1&-2\\1&3&6&1&3\\-7&3&2&-7&3\end{matrix}\right)$$

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

$$-3\times 2-2\times 6\left(-7\right)+5\times 3=93$$

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

$$-7\times 3\times 5+3\times 6\left(-1\right)+2\left(-2\right)=-127$$

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

$$93-\left(-127\right)$$

Subtract -127 from 93.

$$220$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}-1&-2&5\\1&3&6\\-7&3&2\end{matrix}\right))$$

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.

$$-det(\left(\begin{matrix}3&6\\3&2\end{matrix}\right))-\left(-2det(\left(\begin{matrix}1&6\\-7&2\end{matrix}\right))\right)+5det(\left(\begin{matrix}1&3\\-7&3\end{matrix}\right))$$

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

$$-\left(3\times 2-3\times 6\right)-\left(-2\left(2-\left(-7\times 6\right)\right)\right)+5\left(3-\left(-7\times 3\right)\right)$$

Simplify.

$$-\left(-12\right)-\left(-2\times 44\right)+5\times 24$$

Add the terms to obtain the final result.

$$220$$

Factor

$$2^{2}\times 5\times 11$$