Solve {rrr}{5}&{-1}&{-6}{-2}&{5}&{3}{3}&{4}&{2}

$$\left| \begin{array} { r r r } { 5 } & { - 1 } & { - 6 } \\ { - 2 } & { 5 } & { 3 } \\ { 3 } & { 4 } & { 2 } \end{array} \right|$$

Evaluate

$$115$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}5&-1&-6\\-2&5&3\\3&4&2\end{matrix}\right))$$

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

$$\left(\begin{matrix}5&-1&-6&5&-1\\-2&5&3&-2&5\\3&4&2&3&4\end{matrix}\right)$$

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

$$5\times 5\times 2-3\times 3-6\left(-2\right)\times 4=89$$

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

$$3\times 5\left(-6\right)+4\times 3\times 5+2\left(-2\right)\left(-1\right)=-26$$

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

$$89-\left(-26\right)$$

Subtract -26 from 89.

$$115$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}5&-1&-6\\-2&5&3\\3&4&2\end{matrix}\right))$$

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.

$$5det(\left(\begin{matrix}5&3\\4&2\end{matrix}\right))-\left(-det(\left(\begin{matrix}-2&3\\3&2\end{matrix}\right))\right)-6det(\left(\begin{matrix}-2&5\\3&4\end{matrix}\right))$$

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

$$5\left(5\times 2-4\times 3\right)-\left(-\left(-2\times 2-3\times 3\right)\right)-6\left(-2\times 4-3\times 5\right)$$

Simplify.

$$5\left(-2\right)-\left(-\left(-13\right)\right)-6\left(-23\right)$$

Add the terms to obtain the final result.

$$115$$

Factor

$$5\times 23$$