Solve {ccc}{4}&{4}&{2}{-5}&{3}&{-2}{10}&{-2}&{-3}

$$\left| \begin{array} { c c c } { 4 } & { 4 } & { 2 } \\ { - 5 } & { 3 } & { - 2 } \\ { 10 } & { - 2 } & { - 3 } \end{array} \right|$$

Evaluate

$$-232$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}4&4&2\\-5&3&-2\\10&-2&-3\end{matrix}\right))$$

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

$$\left(\begin{matrix}4&4&2&4&4\\-5&3&-2&-5&3\\10&-2&-3&10&-2\end{matrix}\right)$$

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

$$4\times 3\left(-3\right)+4\left(-2\right)\times 10+2\left(-5\right)\left(-2\right)=-96$$

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

$$10\times 3\times 2-2\left(-2\right)\times 4-3\left(-5\right)\times 4=136$$

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

$$-96-136$$

Subtract 136 from -96.

$$-232$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}4&4&2\\-5&3&-2\\10&-2&-3\end{matrix}\right))$$

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.

$$4det(\left(\begin{matrix}3&-2\\-2&-3\end{matrix}\right))-4det(\left(\begin{matrix}-5&-2\\10&-3\end{matrix}\right))+2det(\left(\begin{matrix}-5&3\\10&-2\end{matrix}\right))$$

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

$$4\left(3\left(-3\right)-\left(-2\left(-2\right)\right)\right)-4\left(-5\left(-3\right)-10\left(-2\right)\right)+2\left(-5\left(-2\right)-10\times 3\right)$$

Simplify.

$$4\left(-13\right)-4\times 35+2\left(-20\right)$$

Add the terms to obtain the final result.

$$-232$$

Factor

$$-232$$