Solve {ccc}{1}&{-2}&{3}{102}&{-199}&{301}{-2}&{-1}&{-1}

$$\left| \begin{array} { c c c } { 1 } & { - 2 } & { 3 } \\ { 102 } & { - 199 } & { 301 } \\ { - 2 } & { - 1 } & { - 1 } \end{array} \right|$$

Evaluate

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Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}1&-2&3\\102&-199&301\\-2&-1&-1\end{matrix}\right))$$

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

$$\left(\begin{matrix}1&-2&3&1&-2\\102&-199&301&102&-199\\-2&-1&-1&-2&-1\end{matrix}\right)$$

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

$$-199\left(-1\right)-2\times 301\left(-2\right)+3\times 102\left(-1\right)=1097$$

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

$$-2\left(-199\right)\times 3-301-102\left(-2\right)=1097$$

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

$$1097-1097$$

Subtract 1097 from 1097.

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Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}1&-2&3\\102&-199&301\\-2&-1&-1\end{matrix}\right))$$

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.

$$det(\left(\begin{matrix}-199&301\\-1&-1\end{matrix}\right))-\left(-2det(\left(\begin{matrix}102&301\\-2&-1\end{matrix}\right))\right)+3det(\left(\begin{matrix}102&-199\\-2&-1\end{matrix}\right))$$

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

$$-199\left(-1\right)-\left(-301\right)-\left(-2\left(102\left(-1\right)-\left(-2\times 301\right)\right)\right)+3\left(102\left(-1\right)-\left(-2\left(-199\right)\right)\right)$$

Simplify.

$$500-\left(-2\times 500\right)+3\left(-500\right)$$

Add the terms to obtain the final result.

$$0$$

Factor

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