Solve {lll}{2001}&{2002}&{2003}{2004}&{2005}&{2006}{2007}&{2008}&{2016}

$$\left| \begin{array} { l l l } { 2001 } & { 2002 } & { 2003 } \\ { 2004 } & { 2005 } & { 2006 } \\ { 2007 } & { 2008 } & { 2016 } \end{array} \right|$$

Evaluate

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Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}2001&2002&2003\\2004&2005&2006\\2007&2008&2016\end{matrix}\right))$$

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

$$\left(\begin{matrix}2001&2002&2003&2001&2002\\2004&2005&2006&2004&2005\\2007&2008&2016&2007&2008\end{matrix}\right)$$

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

$$2001\times 2005\times 2016+2002\times 2006\times 2007+2003\times 2004\times 2008=24208474260$$

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

$$2007\times 2005\times 2003+2008\times 2006\times 2001+2016\times 2004\times 2002=24208474281$$

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

$$24208474260-24208474281$$

Subtract 24208474281 from 24208474260.

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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).

$$det(\left(\begin{matrix}2001&2002&2003\\2004&2005&2006\\2007&2008&2016\end{matrix}\right))$$

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.

$$2001det(\left(\begin{matrix}2005&2006\\2008&2016\end{matrix}\right))-2002det(\left(\begin{matrix}2004&2006\\2007&2016\end{matrix}\right))+2003det(\left(\begin{matrix}2004&2005\\2007&2008\end{matrix}\right))$$

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

$$2001\left(2005\times 2016-2008\times 2006\right)-2002\left(2004\times 2016-2007\times 2006\right)+2003\left(2004\times 2008-2007\times 2005\right)$$

Simplify.

$$2001\times 14032-2002\times 14022+2003\left(-3\right)$$

Add the terms to obtain the final result.

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Factor

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