Solve {ccc}{2}&{-1}&{2}{-9}&{2}&{-11/2}{10}&{-5}&{13/2}

$$\left| \begin{array} { c c c } { 2 } & { - 1 } & { 2 } \\ { - 9 } & { 2 } & { - \frac { 11 } { 2 } } \\ { 10 } & { - 5 } & { \frac { 13 } { 2 } } \end{array} \right|$$

Evaluate

$$\frac{35}{2}=17.5$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}2&-1&2\\-9&2&-\frac{11}{2}\\10&-5&\frac{13}{2}\end{matrix}\right))$$

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

$$\left(\begin{matrix}2&-1&2&2&-1\\-9&2&-\frac{11}{2}&-9&2\\10&-5&\frac{13}{2}&10&-5\end{matrix}\right)$$

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

$$2\times 2\times \frac{13}{2}-\left(-\frac{11}{2}\times 10\right)+2\left(-9\right)\left(-5\right)=171$$

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

$$10\times 2\times 2-5\left(-\frac{11}{2}\right)\times 2+\frac{13}{2}\left(-9\right)\left(-1\right)=\frac{307}{2}$$

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

$$171-\frac{307}{2}$$

Subtract \frac{307}{2} from 171.

$$\frac{35}{2}$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}2&-1&2\\-9&2&-\frac{11}{2}\\10&-5&\frac{13}{2}\end{matrix}\right))$$

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.

$$2det(\left(\begin{matrix}2&-\frac{11}{2}\\-5&\frac{13}{2}\end{matrix}\right))-\left(-det(\left(\begin{matrix}-9&-\frac{11}{2}\\10&\frac{13}{2}\end{matrix}\right))\right)+2det(\left(\begin{matrix}-9&2\\10&-5\end{matrix}\right))$$

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

$$2\left(2\times \frac{13}{2}-\left(-5\left(-\frac{11}{2}\right)\right)\right)-\left(-\left(-9\times \frac{13}{2}-10\left(-\frac{11}{2}\right)\right)\right)+2\left(-9\left(-5\right)-10\times 2\right)$$

Simplify.

$$2\left(-\frac{29}{2}\right)-\left(-\left(-\frac{7}{2}\right)\right)+2\times 25$$

Add the terms to obtain the final result.

$$\frac{35}{2}$$

Factor

$$\frac{5 \cdot 7}{2} = 17\frac{1}{2} = 17.5$$