Solve {lll}{5}&{8}&{11}{8}&{13}&{18}{11}&{18}&{25}

$$\left( \begin{array} { l l l } { 5 } & { 8 } & { 11 } \\ { 8 } & { 13 } & { 18 } \\ { 11 } & { 18 } & { 25 } \end{array} \right)$$

Calculate Determinant

$$0$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}5&8&11\\8&13&18\\11&18&25\end{matrix}\right))$$

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

$$\left(\begin{matrix}5&8&11&5&8\\8&13&18&8&13\\11&18&25&11&18\end{matrix}\right)$$

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

$$5\times 13\times 25+8\times 18\times 11+11\times 8\times 18=4793$$

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

$$11\times 13\times 11+18\times 18\times 5+25\times 8\times 8=4793$$

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

$$4793-4793$$

Subtract 4793 from 4793.

$$0$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}5&8&11\\8&13&18\\11&18&25\end{matrix}\right))$$

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.

$$5det(\left(\begin{matrix}13&18\\18&25\end{matrix}\right))-8det(\left(\begin{matrix}8&18\\11&25\end{matrix}\right))+11det(\left(\begin{matrix}8&13\\11&18\end{matrix}\right))$$

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

$$5\left(13\times 25-18\times 18\right)-8\left(8\times 25-11\times 18\right)+11\left(8\times 18-11\times 13\right)$$

Simplify.

$$5-8\times 2+11$$

Add the terms to obtain the final result.

$$0$$

Evaluate

$$\left(\begin{matrix}5&8&11\\8&13&18\\11&18&25\end{matrix}\right)$$