Solve {ccc}{9}&{-10}&{5}{1}&{-7}&{-1}{-7}&{-9}&{-9}

$$\left| \begin{array} { c c c } { 9 } & { - 10 } & { 5 } \\ { 1 } & { - 7 } & { - 1 } \\ { - 7 } & { - 9 } & { - 9 } \end{array} \right|$$

Evaluate

$$36$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}9&-10&5\\1&-7&-1\\-7&-9&-9\end{matrix}\right))$$

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

$$\left(\begin{matrix}9&-10&5&9&-10\\1&-7&-1&1&-7\\-7&-9&-9&-7&-9\end{matrix}\right)$$

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

$$9\left(-7\right)\left(-9\right)-10\left(-1\right)\left(-7\right)+5\left(-9\right)=452$$

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

$$-7\left(-7\right)\times 5-9\left(-1\right)\times 9-9\left(-10\right)=416$$

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

$$452-416$$

Subtract 416 from 452.

$$36$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}9&-10&5\\1&-7&-1\\-7&-9&-9\end{matrix}\right))$$

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.

$$9det(\left(\begin{matrix}-7&-1\\-9&-9\end{matrix}\right))-\left(-10det(\left(\begin{matrix}1&-1\\-7&-9\end{matrix}\right))\right)+5det(\left(\begin{matrix}1&-7\\-7&-9\end{matrix}\right))$$

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

$$9\left(-7\left(-9\right)-\left(-9\left(-1\right)\right)\right)-\left(-10\left(-9-\left(-7\left(-1\right)\right)\right)\right)+5\left(-9-\left(-7\left(-7\right)\right)\right)$$

Simplify.

$$9\times 54-\left(-10\left(-16\right)\right)+5\left(-58\right)$$

Add the terms to obtain the final result.

$$36$$

Factor

$$2^{2}\times 3^{2}$$