Solve {lll}{1}&{-2}&{-4}{4}&{3}&{11}{2}&{-2}&{-11}

$$\left| \begin{array} { l l l } { 1 } & { - 2 } & { - 4 } \\ { 4 } & { 3 } & { 11 } \\ { 2 } & { - 2 } & { - 11 } \end{array} \right|$$

Evaluate

$$-87$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}1&-2&-4\\4&3&11\\2&-2&-11\end{matrix}\right))$$

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

$$\left(\begin{matrix}1&-2&-4&1&-2\\4&3&11&4&3\\2&-2&-11&2&-2\end{matrix}\right)$$

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

$$3\left(-11\right)-2\times 11\times 2-4\times 4\left(-2\right)=-45$$

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

$$2\times 3\left(-4\right)-2\times 11-11\times 4\left(-2\right)=42$$

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

$$-45-42$$

Subtract 42 from -45.

$$-87$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}1&-2&-4\\4&3&11\\2&-2&-11\end{matrix}\right))$$

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.

$$det(\left(\begin{matrix}3&11\\-2&-11\end{matrix}\right))-\left(-2det(\left(\begin{matrix}4&11\\2&-11\end{matrix}\right))\right)-4det(\left(\begin{matrix}4&3\\2&-2\end{matrix}\right))$$

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

$$3\left(-11\right)-\left(-2\times 11\right)-\left(-2\left(4\left(-11\right)-2\times 11\right)\right)-4\left(4\left(-2\right)-2\times 3\right)$$

Simplify.

$$-11-\left(-2\left(-66\right)\right)-4\left(-14\right)$$

Add the terms to obtain the final result.

$$-87$$

Factor

$$-87$$