Solve {ccc}{-2}&{0}&{1}{50}&{200}&{299}{500}&{200}&{300}

$$\left| \begin{array} { c c c } { - 2 } & { 0 } & { 1 } \\ { 50 } & { 200 } & { 299 } \\ { 500 } & { 200 } & { 300 } \end{array} \right|$$

Evaluate

$$-90400$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}-2&0&1\\50&200&299\\500&200&300\end{matrix}\right))$$

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

$$\left(\begin{matrix}-2&0&1&-2&0\\50&200&299&50&200\\500&200&300&500&200\end{matrix}\right)$$

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

$$-2\times 200\times 300+50\times 200=-110000$$

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

$$500\times 200+200\times 299\left(-2\right)=-19600$$

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

$$-110000-\left(-19600\right)$$

Subtract -19600 from -110000.

$$-90400$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}-2&0&1\\50&200&299\\500&200&300\end{matrix}\right))$$

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.

$$-2det(\left(\begin{matrix}200&299\\200&300\end{matrix}\right))+det(\left(\begin{matrix}50&200\\500&200\end{matrix}\right))$$

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

$$-2\left(200\times 300-200\times 299\right)+50\times 200-500\times 200$$

Simplify.

$$-2\times 200-90000$$

Add the terms to obtain the final result.

$$-90400$$

Factor

$$-90400$$