Solve {rrr}{15}&{-6}&{4}{19}&{4}&{-3}{46}&{1}&{6}

$$\left| \begin{array} { r r r } { 15 } & { - 6 } & { 4 } \\ { 19 } & { 4 } & { - 3 } \\ { 46 } & { 1 } & { 6 } \end{array} \right|$$

Evaluate

$$1257$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}15&-6&4\\19&4&-3\\46&1&6\end{matrix}\right))$$

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

$$\left(\begin{matrix}15&-6&4&15&-6\\19&4&-3&19&4\\46&1&6&46&1\end{matrix}\right)$$

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

$$15\times 4\times 6-6\left(-3\right)\times 46+4\times 19=1264$$

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

$$46\times 4\times 4-3\times 15+6\times 19\left(-6\right)=7$$

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

$$1264-7$$

Subtract 7 from 1264.

$$1257$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}15&-6&4\\19&4&-3\\46&1&6\end{matrix}\right))$$

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.

$$15det(\left(\begin{matrix}4&-3\\1&6\end{matrix}\right))-\left(-6det(\left(\begin{matrix}19&-3\\46&6\end{matrix}\right))\right)+4det(\left(\begin{matrix}19&4\\46&1\end{matrix}\right))$$

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

$$15\left(4\times 6-\left(-3\right)\right)-\left(-6\left(19\times 6-46\left(-3\right)\right)\right)+4\left(19-46\times 4\right)$$

Simplify.

$$15\times 27-\left(-6\times 252\right)+4\left(-165\right)$$

Add the terms to obtain the final result.

$$1257$$

Factor

$$3\times 419$$