Solve {ccc}{3/13}&{-5/13}&{4/13}{-4/13}&{11/13}&{-1/13}{1/13}&{71/3}&{-3/13}

$$\left( \begin{array} { c c c } { 3 / 13 } & { - 5 / 13 } & { 4 / 13 } \\ { - 4 / 13 } & { 11 / 13 } & { - 1 / 13 } \\ { 1 / 13 } & { 71 / 3 } & { - 3 / 13 } \end{array} \right)$$

Calculate Determinant

$$-\frac{941}{507} = -1\frac{434}{507} = -1.856015779092702$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}\frac{3}{13}&-\frac{5}{13}&\frac{4}{13}\\-\frac{4}{13}&\frac{11}{13}&-\frac{1}{13}\\\frac{1}{13}&\frac{71}{3}&-\frac{3}{13}\end{matrix}\right))$$

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

$$\left(\begin{matrix}\frac{3}{13}&-\frac{5}{13}&\frac{4}{13}&\frac{3}{13}&-\frac{5}{13}\\-\frac{4}{13}&\frac{11}{13}&-\frac{1}{13}&-\frac{4}{13}&\frac{11}{13}\\\frac{1}{13}&\frac{71}{3}&-\frac{3}{13}&\frac{1}{13}&\frac{71}{3}\end{matrix}\right)$$

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

$$\frac{3}{13}\times \frac{11}{13}\left(-\frac{3}{13}\right)-\frac{5}{13}\left(-\frac{1}{13}\right)\times \frac{1}{13}+\frac{4}{13}\left(-\frac{4}{13}\right)\times \frac{71}{3}=-\frac{15050}{6591}$$

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

$$\frac{1}{13}\times \frac{11}{13}\times \frac{4}{13}+\frac{71}{3}\left(-\frac{1}{13}\right)\times \frac{3}{13}-\frac{3}{13}\left(-\frac{4}{13}\right)\left(-\frac{5}{13}\right)=-\frac{939}{2197}$$

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

$$-\frac{15050}{6591}-\left(-\frac{939}{2197}\right)$$

Subtract -\frac{939}{2197} from -\frac{15050}{6591} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-\frac{941}{507}$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}\frac{3}{13}&-\frac{5}{13}&\frac{4}{13}\\-\frac{4}{13}&\frac{11}{13}&-\frac{1}{13}\\\frac{1}{13}&\frac{71}{3}&-\frac{3}{13}\end{matrix}\right))$$

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.

$$\frac{3}{13}det(\left(\begin{matrix}\frac{11}{13}&-\frac{1}{13}\\\frac{71}{3}&-\frac{3}{13}\end{matrix}\right))-\left(-\frac{5}{13}det(\left(\begin{matrix}-\frac{4}{13}&-\frac{1}{13}\\\frac{1}{13}&-\frac{3}{13}\end{matrix}\right))\right)+\frac{4}{13}det(\left(\begin{matrix}-\frac{4}{13}&\frac{11}{13}\\\frac{1}{13}&\frac{71}{3}\end{matrix}\right))$$

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

$$\frac{3}{13}\left(\frac{11}{13}\left(-\frac{3}{13}\right)-\frac{71}{3}\left(-\frac{1}{13}\right)\right)-\left(-\frac{5}{13}\left(-\frac{4}{13}\left(-\frac{3}{13}\right)-\frac{1}{13}\left(-\frac{1}{13}\right)\right)\right)+\frac{4}{13}\left(-\frac{4}{13}\times \frac{71}{3}-\frac{1}{13}\times \frac{11}{13}\right)$$

Simplify.

$$\frac{3}{13}\times \frac{824}{507}-\left(-\frac{5}{13}\times \frac{1}{13}\right)+\frac{4}{13}\left(-\frac{3725}{507}\right)$$

Add the terms to obtain the final result.

$$-\frac{941}{507}$$

Evaluate

$$\left(\begin{matrix}\frac{3}{13}&-\frac{5}{13}&\frac{4}{13}\\-\frac{4}{13}&\frac{11}{13}&-\frac{1}{13}\\\frac{1}{13}&\frac{71}{3}&-\frac{3}{13}\end{matrix}\right)$$