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

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

Evaluate

$$65$$

Steps Using Sarrus' Rule

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

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

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

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

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

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

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

$$3\left(-5\right)\times 3+4\times 5\left(-2\right)-\left(-5\times 5\right)=-60$$

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

$$5-\left(-60\right)$$

Subtract -60 from 5.

$$65$$

Steps Using Expansion by Minors

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

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

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.

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

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

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

Simplify.

$$-2\left(-15\right)-5\left(-10\right)+3\left(-5\right)$$

Add the terms to obtain the final result.

$$65$$

Factor

$$5\times 13$$