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

$$\left( \begin{array} { c c c } { 2 } & { 4 } & { 4 } \\ { 4 } & { - 2 } & { - 4 } \\ { 5 } & { 4 } & { 4 } \end{array} \right)$$

Calculate Determinant

$$-24$$

Steps Using Sarrus' Rule

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

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

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

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

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

$$2\left(-2\right)\times 4+4\left(-4\right)\times 5+4\times 4\times 4=-32$$

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

$$5\left(-2\right)\times 4+4\left(-4\right)\times 2+4\times 4\times 4=-8$$

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

$$-32-\left(-8\right)$$

Subtract -8 from -32.

$$-24$$

Steps Using Expansion by Minors

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

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

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.

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

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

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

Simplify.

$$2\times 8-4\times 36+4\times 26$$

Add the terms to obtain the final result.

$$-24$$

Evaluate

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