Solve {lll}{103}&{100}&{204}{199}&{200}&{395}{301}&{300}&{600}=

$$\left| \begin{array} { l l l } { 103 } & { 100 } & { 204 } \\ { 199 } & { 200 } & { 395 } \\ { 301 } & { 300 } & { 600 } \end{array} \right| =$$

Evaluate

$$2000$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}103&100&204\\199&200&395\\301&300&600\end{matrix}\right))$$

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

$$\left(\begin{matrix}103&100&204&103&100\\199&200&395&199&200\\301&300&600&301&300\end{matrix}\right)$$

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

$$103\times 200\times 600+100\times 395\times 301+204\times 199\times 300=36428300$$

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

$$301\times 200\times 204+300\times 395\times 103+600\times 199\times 100=36426300$$

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

$$36428300-36426300$$

Subtract 36426300 from 36428300.

$$2000$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}103&100&204\\199&200&395\\301&300&600\end{matrix}\right))$$

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.

$$103det(\left(\begin{matrix}200&395\\300&600\end{matrix}\right))-100det(\left(\begin{matrix}199&395\\301&600\end{matrix}\right))+204det(\left(\begin{matrix}199&200\\301&300\end{matrix}\right))$$

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

$$103\left(200\times 600-300\times 395\right)-100\left(199\times 600-301\times 395\right)+204\left(199\times 300-301\times 200\right)$$

Simplify.

$$103\times 1500-100\times 505+204\left(-500\right)$$

Add the terms to obtain the final result.

$$2000$$

Factor

$$2^{4}\times 5^{3}$$