Solve {ccc}{-5}&{-5}&{3}{1}&{-2}&{-2}{1}&{-4}&{4}

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

Evaluate

$$104$$

Steps Using Sarrus' Rule

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

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

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

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

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

$$-5\left(-2\right)\times 4-5\left(-2\right)+3\left(-4\right)=38$$

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

$$-2\times 3-4\left(-2\right)\left(-5\right)+4\left(-5\right)=-66$$

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

$$38-\left(-66\right)$$

Subtract -66 from 38.

$$104$$

Steps Using Expansion by Minors

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

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

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.

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

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

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

Simplify.

$$-5\left(-16\right)-\left(-5\times 6\right)+3\left(-2\right)$$

Add the terms to obtain the final result.

$$104$$

Factor

$$2^{3}\times 13$$