Solve {lll}{100}&{205}&{105}{200}&{408}&{207}{300}&{608}&{310}

$$\left| \begin{array} { l l l } { 100 } & { 205 } & { 105 } \\ { 200 } & { 408 } & { 207 } \\ { 300 } & { 608 } & { 310 } \end{array} \right|$$

Evaluate

$$-1100$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}100&205&105\\200&408&207\\300&608&310\end{matrix}\right))$$

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

$$\left(\begin{matrix}100&205&105&100&205\\200&408&207&200&408\\300&608&310&300&608\end{matrix}\right)$$

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

$$100\times 408\times 310+205\times 207\times 300+105\times 200\times 608=38146500$$

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

$$300\times 408\times 105+608\times 207\times 100+310\times 200\times 205=38147600$$

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

$$38146500-38147600$$

Subtract 38147600 from 38146500.

$$-1100$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}100&205&105\\200&408&207\\300&608&310\end{matrix}\right))$$

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.

$$100det(\left(\begin{matrix}408&207\\608&310\end{matrix}\right))-205det(\left(\begin{matrix}200&207\\300&310\end{matrix}\right))+105det(\left(\begin{matrix}200&408\\300&608\end{matrix}\right))$$

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

$$100\left(408\times 310-608\times 207\right)-205\left(200\times 310-300\times 207\right)+105\left(200\times 608-300\times 408\right)$$

Simplify.

$$100\times 624-205\left(-100\right)+105\left(-800\right)$$

Add the terms to obtain the final result.

$$-1100$$

Factor

$$-1100$$