Solve {ccc}{5}&{-7}&{1}{6}&{-8}&{-1}{3}&{2}&{-6}

$$\left| \begin{array} { c c c } { 5 } & { - 7 } & { 1 } \\ { 6 } & { - 8 } & { - 1 } \\ { 3 } & { 2 } & { - 6 } \end{array} \right|$$

Evaluate

$$55$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}5&-7&1\\6&-8&-1\\3&2&-6\end{matrix}\right))$$

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

$$\left(\begin{matrix}5&-7&1&5&-7\\6&-8&-1&6&-8\\3&2&-6&3&2\end{matrix}\right)$$

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

$$5\left(-8\right)\left(-6\right)-7\left(-1\right)\times 3+6\times 2=273$$

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

$$3\left(-8\right)+2\left(-1\right)\times 5-6\times 6\left(-7\right)=218$$

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

$$273-218$$

Subtract 218 from 273.

$$55$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}5&-7&1\\6&-8&-1\\3&2&-6\end{matrix}\right))$$

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.

$$5det(\left(\begin{matrix}-8&-1\\2&-6\end{matrix}\right))-\left(-7det(\left(\begin{matrix}6&-1\\3&-6\end{matrix}\right))\right)+det(\left(\begin{matrix}6&-8\\3&2\end{matrix}\right))$$

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

$$5\left(-8\left(-6\right)-2\left(-1\right)\right)-\left(-7\left(6\left(-6\right)-3\left(-1\right)\right)\right)+6\times 2-3\left(-8\right)$$

Simplify.

$$5\times 50-\left(-7\left(-33\right)\right)+36$$

Add the terms to obtain the final result.

$$55$$

Factor

$$5\times 11$$