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

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

Evaluate

$$-4$$

Steps Using Sarrus' Rule

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

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

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

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

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

$$-\left(-3\right)-2\left(-2\right)\left(-2\right)-\left(-3\right)=-2$$

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

$$-2\left(-1\right)-3\left(-2\right)\left(-1\right)-3\left(-2\right)=2$$

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

$$-2-2$$

Subtract 2 from -2.

$$-4$$

Steps Using Expansion by Minors

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

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

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.

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

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

$$-\left(-3-\left(-3\left(-2\right)\right)\right)-\left(-2\left(-3-\left(-2\left(-2\right)\right)\right)\right)-\left(-3-\left(-2\right)\right)$$

Simplify.

$$-\left(-9\right)-\left(-2\left(-7\right)\right)-\left(-1\right)$$

Add the terms to obtain the final result.

$$-4$$

Factor

$$-4$$