Solve {lll}{11}&{21}&{31}{12}&{22}&{32}{13}&{23}&{33}

$$\left| \begin{array} { l l l } { 11 } & { 21 } & { 31 } \\ { 12 } & { 22 } & { 32 } \\ { 13 } & { 23 } & { 33 } \end{array} \right|$$

Evaluate

$$0$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}11&21&31\\12&22&32\\13&23&33\end{matrix}\right))$$

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

$$\left(\begin{matrix}11&21&31&11&21\\12&22&32&12&22\\13&23&33&13&23\end{matrix}\right)$$

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

$$11\times 22\times 33+21\times 32\times 13+31\times 12\times 23=25278$$

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

$$13\times 22\times 31+23\times 32\times 11+33\times 12\times 21=25278$$

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

$$25278-25278$$

Subtract 25278 from 25278.

$$0$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}11&21&31\\12&22&32\\13&23&33\end{matrix}\right))$$

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.

$$11det(\left(\begin{matrix}22&32\\23&33\end{matrix}\right))-21det(\left(\begin{matrix}12&32\\13&33\end{matrix}\right))+31det(\left(\begin{matrix}12&22\\13&23\end{matrix}\right))$$

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

$$11\left(22\times 33-23\times 32\right)-21\left(12\times 33-13\times 32\right)+31\left(12\times 23-13\times 22\right)$$

Simplify.

$$11\left(-10\right)-21\left(-20\right)+31\left(-10\right)$$

Add the terms to obtain the final result.

$$0$$

Factor

$$0$$