Solve {lll}{11}&{21}&{31}{21}&{31}&{4}{31}&{41}&{51}

$$\left| \begin{array} { l l l } { 11 } & { 21 } & { 31 } \\ { 21 } & { 31 } & { 4 } \\ { 31 } & { 41 } & { 51 } \end{array} \right|$$

Evaluate

$$-7400$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}11&21&31\\21&31&4\\31&41&51\end{matrix}\right))$$

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

$$\left(\begin{matrix}11&21&31&11&21\\21&31&4&21&31\\31&41&51&31&41\end{matrix}\right)$$

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

$$11\times 31\times 51+21\times 4\times 31+31\times 21\times 41=46686$$

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

$$31\times 31\times 31+41\times 4\times 11+51\times 21\times 21=54086$$

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

$$46686-54086$$

Subtract 54086 from 46686.

$$-7400$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}11&21&31\\21&31&4\\31&41&51\end{matrix}\right))$$

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.

$$11det(\left(\begin{matrix}31&4\\41&51\end{matrix}\right))-21det(\left(\begin{matrix}21&4\\31&51\end{matrix}\right))+31det(\left(\begin{matrix}21&31\\31&41\end{matrix}\right))$$

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

$$11\left(31\times 51-41\times 4\right)-21\left(21\times 51-31\times 4\right)+31\left(21\times 41-31\times 31\right)$$

Simplify.

$$11\times 1417-21\times 947+31\left(-100\right)$$

Add the terms to obtain the final result.

$$-7400$$

Factor

$$-7400$$