Solve {ccc}{1}&{-16}&{19}{7}&{-6}&{13}{3}&{6}&{4}

$$\left| \begin{array} { c c c } { 1 } & { - 16 } & { 19 } \\ { 7 } & { - 6 } & { 13 } \\ { 3 } & { 6 } & { 4 } \end{array} \right|$$

Evaluate

$$862$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}1&-16&19\\7&-6&13\\3&6&4\end{matrix}\right))$$

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

$$\left(\begin{matrix}1&-16&19&1&-16\\7&-6&13&7&-6\\3&6&4&3&6\end{matrix}\right)$$

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

$$-6\times 4-16\times 13\times 3+19\times 7\times 6=150$$

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

$$3\left(-6\right)\times 19+6\times 13+4\times 7\left(-16\right)=-712$$

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

$$150-\left(-712\right)$$

Subtract -712 from 150.

$$862$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}1&-16&19\\7&-6&13\\3&6&4\end{matrix}\right))$$

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.

$$det(\left(\begin{matrix}-6&13\\6&4\end{matrix}\right))-\left(-16det(\left(\begin{matrix}7&13\\3&4\end{matrix}\right))\right)+19det(\left(\begin{matrix}7&-6\\3&6\end{matrix}\right))$$

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

$$-6\times 4-6\times 13-\left(-16\left(7\times 4-3\times 13\right)\right)+19\left(7\times 6-3\left(-6\right)\right)$$

Simplify.

$$-102-\left(-16\left(-11\right)\right)+19\times 60$$

Add the terms to obtain the final result.

$$862$$

Factor

$$2\times 431$$