Solve {ccc}{197}&{100}&{1}{201}&{200}&{-1}{199}&{300}&{0}

$$\left| \begin{array} { c c c } { 197 } & { 100 } & { 1 } \\ { 201 } & { 200 } & { - 1 } \\ { 199 } & { 300 } & { 0 } \end{array} \right|$$

Evaluate

$$59700$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}197&100&1\\201&200&-1\\199&300&0\end{matrix}\right))$$

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

$$\left(\begin{matrix}197&100&1&197&100\\201&200&-1&201&200\\199&300&0&199&300\end{matrix}\right)$$

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

$$100\left(-1\right)\times 199+201\times 300=40400$$

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

$$199\times 200+300\left(-1\right)\times 197=-19300$$

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

$$40400-\left(-19300\right)$$

Subtract -19300 from 40400.

$$59700$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}197&100&1\\201&200&-1\\199&300&0\end{matrix}\right))$$

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.

$$197det(\left(\begin{matrix}200&-1\\300&0\end{matrix}\right))-100det(\left(\begin{matrix}201&-1\\199&0\end{matrix}\right))+det(\left(\begin{matrix}201&200\\199&300\end{matrix}\right))$$

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

$$197\left(-300\left(-1\right)\right)-100\left(-199\left(-1\right)\right)+201\times 300-199\times 200$$

Simplify.

$$197\times 300-100\times 199+20500$$

Add the terms to obtain the final result.

$$59700$$

Factor

$$2^{2}\times 3\times 5^{2}\times 199$$