Solve {ccc}{2}&{3}&{5}{400}&{600}&{100}{48}&{47}&{18}

$$\left| \begin{array} { c c c } { 2 } & { 3 } & { 5 } \\ { 400 } & { 600 } & { 100 } \\ { 48 } & { 47 } & { 18 } \end{array} \right|$$

Evaluate

$$-45000$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}2&3&5\\400&600&100\\48&47&18\end{matrix}\right))$$

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

$$\left(\begin{matrix}2&3&5&2&3\\400&600&100&400&600\\48&47&18&48&47\end{matrix}\right)$$

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

$$2\times 600\times 18+3\times 100\times 48+5\times 400\times 47=130000$$

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

$$48\times 600\times 5+47\times 100\times 2+18\times 400\times 3=175000$$

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

$$130000-175000$$

Subtract 175000 from 130000.

$$-45000$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}2&3&5\\400&600&100\\48&47&18\end{matrix}\right))$$

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.

$$2det(\left(\begin{matrix}600&100\\47&18\end{matrix}\right))-3det(\left(\begin{matrix}400&100\\48&18\end{matrix}\right))+5det(\left(\begin{matrix}400&600\\48&47\end{matrix}\right))$$

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

$$2\left(600\times 18-47\times 100\right)-3\left(400\times 18-48\times 100\right)+5\left(400\times 47-48\times 600\right)$$

Simplify.

$$2\times 6100-3\times 2400+5\left(-10000\right)$$

Add the terms to obtain the final result.

$$-45000$$

Factor

$$-45000$$