Solve {ccc}{9/2}&{7/8}&{17/3}{28/3}&{frac{37}{9}}&{frac{9}{8}}{frac{1}{13}}&{frac{8}{9}}&{frac{2}{7}}

$$\left| \begin{array} { c c c } { \frac { 9 } { 2 } } & { \frac { 7 } { 8 } } & { \frac { 17 } { 3 } } \\ { \frac { 28 } { 3 } } & { \frac { 37 } { 9 } } & { \frac { 9 } { 8 } } \\ { \frac { 1 } { 13 } } & { \frac { 8 } { 9 } } & { \frac { 2 } { 7 } } \end{array} \right|$$

Evaluate

$$\frac{20638057}{471744}\approx 43.748424993$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}\frac{9}{2}&\frac{7}{8}&\frac{17}{3}\\\frac{28}{3}&\frac{37}{9}&\frac{9}{8}\\\frac{1}{13}&\frac{8}{9}&\frac{2}{7}\end{matrix}\right))$$

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

$$\left(\begin{matrix}\frac{9}{2}&\frac{7}{8}&\frac{17}{3}&\frac{9}{2}&\frac{7}{8}\\\frac{28}{3}&\frac{37}{9}&\frac{9}{8}&\frac{28}{3}&\frac{37}{9}\\\frac{1}{13}&\frac{8}{9}&\frac{2}{7}&\frac{1}{13}&\frac{8}{9}\end{matrix}\right)$$

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

$$\frac{9}{2}\times \frac{37}{9}\times \frac{2}{7}+\frac{7}{8}\times \frac{9}{8}\times \frac{1}{13}+\frac{17}{3}\times \frac{28}{3}\times \frac{8}{9}=\frac{24707017}{471744}$$

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

$$\frac{1}{13}\times \frac{37}{9}\times \frac{17}{3}+\frac{8}{9}\times \frac{9}{8}\times \frac{9}{2}+\frac{2}{7}\times \frac{28}{3}\times \frac{7}{8}=\frac{6055}{702}$$

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

$$\frac{24707017}{471744}-\frac{6055}{702}$$

Subtract \frac{6055}{702} from \frac{24707017}{471744} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$\frac{20638057}{471744}$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}\frac{9}{2}&\frac{7}{8}&\frac{17}{3}\\\frac{28}{3}&\frac{37}{9}&\frac{9}{8}\\\frac{1}{13}&\frac{8}{9}&\frac{2}{7}\end{matrix}\right))$$

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.

$$\frac{9}{2}det(\left(\begin{matrix}\frac{37}{9}&\frac{9}{8}\\\frac{8}{9}&\frac{2}{7}\end{matrix}\right))-\frac{7}{8}det(\left(\begin{matrix}\frac{28}{3}&\frac{9}{8}\\\frac{1}{13}&\frac{2}{7}\end{matrix}\right))+\frac{17}{3}det(\left(\begin{matrix}\frac{28}{3}&\frac{37}{9}\\\frac{1}{13}&\frac{8}{9}\end{matrix}\right))$$

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

$$\frac{9}{2}\left(\frac{37}{9}\times \frac{2}{7}-\frac{8}{9}\times \frac{9}{8}\right)-\frac{7}{8}\left(\frac{28}{3}\times \frac{2}{7}-\frac{1}{13}\times \frac{9}{8}\right)+\frac{17}{3}\left(\frac{28}{3}\times \frac{8}{9}-\frac{1}{13}\times \frac{37}{9}\right)$$

Simplify.

$$\frac{9}{2}\times \frac{11}{63}-\frac{7}{8}\times \frac{805}{312}+\frac{17}{3}\times \frac{2801}{351}$$

Add the terms to obtain the final result.

$$\frac{20638057}{471744}$$

Factor

$$\frac{11 \cdot 1876187}{2 ^ {6} \cdot 3 ^ {4} \cdot 7 \cdot 13} = 43\frac{353065}{471744} = 43.74842499321666$$