Solve {c|c|c}hline6&{11}&{25}hline8&{6}&{16}hline12&{5}&{?}hline

$$\left. \begin{array} { c | c | c } \hline 6 & { 11 } & { 25 } \\ \hline 8 & { 6 } & { 16 } \\ \hline 12 & { 5 } & { ? } \\ \hline \end{array} \right.$$

Calculate Determinant

$$832$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}6&11&25\\8&6&16\\12&5&0\end{matrix}\right))$$

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

$$\left(\begin{matrix}6&11&25&6&11\\8&6&16&8&6\\12&5&0&12&5\end{matrix}\right)$$

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

$$11\times 16\times 12+25\times 8\times 5=3112$$

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

$$12\times 6\times 25+5\times 16\times 6=2280$$

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

$$3112-2280$$

Subtract 2280 from 3112.

$$832$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}6&11&25\\8&6&16\\12&5&0\end{matrix}\right))$$

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.

$$6det(\left(\begin{matrix}6&16\\5&0\end{matrix}\right))-11det(\left(\begin{matrix}8&16\\12&0\end{matrix}\right))+25det(\left(\begin{matrix}8&6\\12&5\end{matrix}\right))$$

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

$$6\left(-5\times 16\right)-11\left(-12\times 16\right)+25\left(8\times 5-12\times 6\right)$$

Simplify.

$$6\left(-80\right)-11\left(-192\right)+25\left(-32\right)$$

Add the terms to obtain the final result.

$$832$$

Evaluate

$$\left(\begin{matrix}6&11&25\\8&6&16\\12&5&0\end{matrix}\right)$$