Solve {ccc}{1}&{-18}&{-11}{3}&{6}&{-4}{13}&{8}&{3}

$$\left| \begin{array} { c c c } { 1 } & { - 18 } & { - 11 } \\ { 3 } & { 6 } & { - 4 } \\ { 13 } & { 8 } & { 3 } \end{array} \right|$$

Evaluate

$$1742$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}1&-18&-11\\3&6&-4\\13&8&3\end{matrix}\right))$$

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

$$\left(\begin{matrix}1&-18&-11&1&-18\\3&6&-4&3&6\\13&8&3&13&8\end{matrix}\right)$$

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

$$6\times 3-18\left(-4\right)\times 13-11\times 3\times 8=690$$

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

$$13\times 6\left(-11\right)+8\left(-4\right)+3\times 3\left(-18\right)=-1052$$

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

$$690-\left(-1052\right)$$

Subtract -1052 from 690.

$$1742$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}1&-18&-11\\3&6&-4\\13&8&3\end{matrix}\right))$$

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.

$$det(\left(\begin{matrix}6&-4\\8&3\end{matrix}\right))-\left(-18det(\left(\begin{matrix}3&-4\\13&3\end{matrix}\right))\right)-11det(\left(\begin{matrix}3&6\\13&8\end{matrix}\right))$$

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

$$6\times 3-8\left(-4\right)-\left(-18\left(3\times 3-13\left(-4\right)\right)\right)-11\left(3\times 8-13\times 6\right)$$

Simplify.

$$50-\left(-18\times 61\right)-11\left(-54\right)$$

Add the terms to obtain the final result.

$$1742$$

Factor

$$2\times 13\times 67$$