Solve {ccc}{2}&{0}&{201}{1}&{-4}&{99}{-1}&{8}&{-97}

$$\left| \begin{array} { c c c } { 2 } & { 0 } & { 201 } \\ { 1 } & { - 4 } & { 99 } \\ { - 1 } & { 8 } & { - 97 } \end{array} \right|$$

Evaluate

$$-4$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}2&0&201\\1&-4&99\\-1&8&-97\end{matrix}\right))$$

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

$$\left(\begin{matrix}2&0&201&2&0\\1&-4&99&1&-4\\-1&8&-97&-1&8\end{matrix}\right)$$

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

$$2\left(-4\right)\left(-97\right)+201\times 8=2384$$

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

$$-\left(-4\right)\times 201+8\times 99\times 2=2388$$

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

$$2384-2388$$

Subtract 2388 from 2384.

$$-4$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}2&0&201\\1&-4&99\\-1&8&-97\end{matrix}\right))$$

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.

$$2det(\left(\begin{matrix}-4&99\\8&-97\end{matrix}\right))+201det(\left(\begin{matrix}1&-4\\-1&8\end{matrix}\right))$$

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

$$2\left(-4\left(-97\right)-8\times 99\right)+201\left(8-\left(-\left(-4\right)\right)\right)$$

Simplify.

$$2\left(-404\right)+201\times 4$$

Add the terms to obtain the final result.

$$-4$$

Factor

$$-4$$