Solve {rrr}{-3}&{-3}&{-11}{2}&{-3}&{8}{1}&{-4}&{-7}

$$\left| \begin{array} { r r r } { - 3 } & { - 3 } & { - 11 } \\ { 2 } & { - 3 } & { 8 } \\ { 1 } & { - 4 } & { - 7 } \end{array} \right|$$

Evaluate

$$-170$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}-3&-3&-11\\2&-3&8\\1&-4&-7\end{matrix}\right))$$

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

$$\left(\begin{matrix}-3&-3&-11&-3&-3\\2&-3&8&2&-3\\1&-4&-7&1&-4\end{matrix}\right)$$

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

$$-3\left(-3\right)\left(-7\right)-3\times 8-11\times 2\left(-4\right)=1$$

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

$$-3\left(-11\right)-4\times 8\left(-3\right)-7\times 2\left(-3\right)=171$$

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

$$1-171$$

Subtract 171 from 1.

$$-170$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}-3&-3&-11\\2&-3&8\\1&-4&-7\end{matrix}\right))$$

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.

$$-3det(\left(\begin{matrix}-3&8\\-4&-7\end{matrix}\right))-\left(-3det(\left(\begin{matrix}2&8\\1&-7\end{matrix}\right))\right)-11det(\left(\begin{matrix}2&-3\\1&-4\end{matrix}\right))$$

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

$$-3\left(-3\left(-7\right)-\left(-4\times 8\right)\right)-\left(-3\left(2\left(-7\right)-8\right)\right)-11\left(2\left(-4\right)-\left(-3\right)\right)$$

Simplify.

$$-3\times 53-\left(-3\left(-22\right)\right)-11\left(-5\right)$$

Add the terms to obtain the final result.

$$-170$$

Factor

$$-170$$