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

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

Evaluate

$$19$$

Steps Using Sarrus' Rule

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

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

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

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

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

$$3\left(-1\right)\times 6-5\times 2\left(-2\right)-\left(-7\right)=9$$

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

$$-2\left(-1\right)-7\times 2\times 3+6\left(-1\right)\left(-5\right)=-10$$

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

$$9-\left(-10\right)$$

Subtract -10 from 9.

$$19$$

Steps Using Expansion by Minors

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

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

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.

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

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

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

Simplify.

$$3\times 8-\left(-5\left(-2\right)\right)+5$$

Add the terms to obtain the final result.

$$19$$

Factor

$$19$$