Solve {ccc}{1}&{2}&{-2}{2}&{-2}&{-1}{2}&{1}&{2}

$$\left( \begin{array} { c c c } { 1 } & { 2 } & { - 2 } \\ { 2 } & { - 2 } & { - 1 } \\ { 2 } & { 1 } & { 2 } \end{array} \right)$$

Calculate Determinant

$$-27$$

Steps Using Sarrus' Rule

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

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

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

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

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

$$-2\times 2+2\left(-1\right)\times 2-2\times 2=-12$$

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

$$2\left(-2\right)\left(-2\right)-1+2\times 2\times 2=15$$

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

$$-12-15$$

Subtract 15 from -12.

$$-27$$

Steps Using Expansion by Minors

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

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

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.

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

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

$$-2\times 2-\left(-1\right)-2\left(2\times 2-2\left(-1\right)\right)-2\left(2-2\left(-2\right)\right)$$

Simplify.

$$-3-2\times 6-2\times 6$$

Add the terms to obtain the final result.

$$-27$$

Evaluate

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