Solve {ccc}hline1,519&{1,009}&{607}{921/2,440}&{644/1,653}&{464}hline&{1,071}

$$\left. \begin{array} { c c c } \hline 1,519 & { 1,009 } & { 607 } \\ { \frac { 921 } { 2,440 } } & { \frac { 644 } { 1,653 } } & { 464 } \\ \hline & { 1,071 } \end{array} \right.$$

Calculate Determinant

$$\frac{1865304636363}{7625000} = 244630\frac{886363}{7625000} = 244630.11624432786$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}1,519&1,009&607\\\frac{23025}{61}&\frac{644000}{1653}&464\\0&1,071&0\end{matrix}\right))$$

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

$$\left(\begin{matrix}1,519&1,009&607&1,519&1,009\\\frac{23025}{61}&\frac{644000}{1653}&464&\frac{23025}{61}&\frac{644000}{1653}\\0&1,071&0&0&1,071\end{matrix}\right)$$

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

$$607\times \frac{23025}{61}\times 1,071=\frac{598739337}{2440}$$

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

$$1,071\times 464\times 1,519=754,857936$$

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

$$\frac{598739337}{2440}-754,857936$$

Subtract 754,857936 from \frac{598739337}{2440} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$\frac{1865304636363}{7625000}$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}1,519&1,009&607\\\frac{23025}{61}&\frac{644000}{1653}&464\\0&1,071&0\end{matrix}\right))$$

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.

$$1,519det(\left(\begin{matrix}\frac{644000}{1653}&464\\1,071&0\end{matrix}\right))-1,009det(\left(\begin{matrix}\frac{23025}{61}&464\\0&0\end{matrix}\right))+607det(\left(\begin{matrix}\frac{23025}{61}&\frac{644000}{1653}\\0&1,071\end{matrix}\right))$$

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

$$1,519\left(-1,071\times 464\right)+607\times \frac{23025}{61}\times 1,071$$

Simplify.

$$1,519\left(-496,944\right)+607\times \frac{986391}{2440}$$

Add the terms to obtain the final result.

$$\frac{1865304636363}{7625000}$$

Evaluate

$$\left(\begin{matrix}1,519&1,009&607\\\frac{23025}{61}&\frac{644000}{1653}&464\\0&1,071&0\end{matrix}\right)$$