Solve {ccc}{5}&{6}&{7}{8}&{9}&{10}{11}&{12}&{13}=

$$\left( \begin{array} { c c c } { 5 } & { 6 } & { 7 } \\ { 8 } & { 9 } & { 10 } \\ { 11 } & { 12 } & { 13 } \end{array} \right) =$$

Calculate Determinant

$$0$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}5&6&7\\8&9&10\\11&12&13\end{matrix}\right))$$

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

$$\left(\begin{matrix}5&6&7&5&6\\8&9&10&8&9\\11&12&13&11&12\end{matrix}\right)$$

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

$$5\times 9\times 13+6\times 10\times 11+7\times 8\times 12=1917$$

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

$$11\times 9\times 7+12\times 10\times 5+13\times 8\times 6=1917$$

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

$$1917-1917$$

Subtract 1917 from 1917.

$$0$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}5&6&7\\8&9&10\\11&12&13\end{matrix}\right))$$

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.

$$5det(\left(\begin{matrix}9&10\\12&13\end{matrix}\right))-6det(\left(\begin{matrix}8&10\\11&13\end{matrix}\right))+7det(\left(\begin{matrix}8&9\\11&12\end{matrix}\right))$$

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

$$5\left(9\times 13-12\times 10\right)-6\left(8\times 13-11\times 10\right)+7\left(8\times 12-11\times 9\right)$$

Simplify.

$$5\left(-3\right)-6\left(-6\right)+7\left(-3\right)$$

Add the terms to obtain the final result.

$$0$$

Evaluate

$$\left(\begin{matrix}5&6&7\\8&9&10\\11&12&13\end{matrix}\right)$$