Solve {ccc}{5}&{1}&{-5}{3}&{-4}&{5}{-4}&{-3}&{6}

$$\left| \begin{array} { c c c } { 5 } & { 1 } & { - 5 } \\ { 3 } & { - 4 } & { 5 } \\ { - 4 } & { - 3 } & { 6 } \end{array} \right|$$

Evaluate

$$42$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}5&1&-5\\3&-4&5\\-4&-3&6\end{matrix}\right))$$

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

$$\left(\begin{matrix}5&1&-5&5&1\\3&-4&5&3&-4\\-4&-3&6&-4&-3\end{matrix}\right)$$

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

$$5\left(-4\right)\times 6+5\left(-4\right)-5\times 3\left(-3\right)=-95$$

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

$$-4\left(-4\right)\left(-5\right)-3\times 5\times 5+6\times 3=-137$$

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

$$-95-\left(-137\right)$$

Subtract -137 from -95.

$$42$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}5&1&-5\\3&-4&5\\-4&-3&6\end{matrix}\right))$$

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.

$$5det(\left(\begin{matrix}-4&5\\-3&6\end{matrix}\right))-det(\left(\begin{matrix}3&5\\-4&6\end{matrix}\right))-5det(\left(\begin{matrix}3&-4\\-4&-3\end{matrix}\right))$$

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

$$5\left(-4\times 6-\left(-3\times 5\right)\right)-\left(3\times 6-\left(-4\times 5\right)\right)-5\left(3\left(-3\right)-\left(-4\left(-4\right)\right)\right)$$

Simplify.

$$5\left(-9\right)-38-5\left(-25\right)$$

Add the terms to obtain the final result.

$$42$$

Factor

$$2\times 3\times 7$$