Solve {lll}{1998}&{1999}&{2000}{2001}&{2002}&{2003}{2004}&{2005}&{2006}

$$\left| \begin{array} { l l l } { 1998 } & { 1999 } & { 2000 } \\ { 2001 } & { 2002 } & { 2003 } \\ { 2004 } & { 2005 } & { 2006 } \end{array} \right|$$

Evaluate

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Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}1998&1999&2000\\2001&2002&2003\\2004&2005&2006\end{matrix}\right))$$

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

$$\left(\begin{matrix}1998&1999&2000&1998&1999\\2001&2002&2003&2001&2002\\2004&2005&2006&2004&2005\end{matrix}\right)$$

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

$$1998\times 2002\times 2006+1999\times 2003\times 2004+2000\times 2001\times 2005=24072011964$$

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

$$2004\times 2002\times 2000+2005\times 2003\times 1998+2006\times 2001\times 1999=24072011964$$

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

$$24072011964-24072011964$$

Subtract 24072011964 from 24072011964.

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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).

$$det(\left(\begin{matrix}1998&1999&2000\\2001&2002&2003\\2004&2005&2006\end{matrix}\right))$$

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.

$$1998det(\left(\begin{matrix}2002&2003\\2005&2006\end{matrix}\right))-1999det(\left(\begin{matrix}2001&2003\\2004&2006\end{matrix}\right))+2000det(\left(\begin{matrix}2001&2002\\2004&2005\end{matrix}\right))$$

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

$$1998\left(2002\times 2006-2005\times 2003\right)-1999\left(2001\times 2006-2004\times 2003\right)+2000\left(2001\times 2005-2004\times 2002\right)$$

Simplify.

$$1998\left(-3\right)-1999\left(-6\right)+2000\left(-3\right)$$

Add the terms to obtain the final result.

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Factor

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