Solve {ccc}{2+8}&{-342}&{399}{-92}&{138}&{-161}{1116}&{93}&{186}

$$\left| \begin{array} { c c c } { 2 + 8 } & { - 342 } & { 399 } \\ { - 92 } & { 138 } & { - 161 } \\ { 1116 } & { 93 } & { 186 } \end{array} \right|$$

Evaluate

$$-8859738$$

Steps Using Sarrus' Rule

Add 2 and 8 to get 10.

$$det(\left(\begin{matrix}10&-342&399\\-92&138&-161\\1116&93&186\end{matrix}\right))$$

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

$$\left(\begin{matrix}10&-342&399&10&-342\\-92&138&-161&-92&138\\1116&93&186&1116&93\end{matrix}\right)$$

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

$$10\times 138\times 186-342\left(-161\right)\times 1116+399\left(-92\right)\times 93=58292028$$

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

$$1116\times 138\times 399+93\left(-161\right)\times 10+186\left(-92\right)\left(-342\right)=67151766$$

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

$$58292028-67151766$$

Subtract 67151766 from 58292028.

$$-8859738$$

Steps Using Expansion by Minors

Add 2 and 8 to get 10.

$$det(\left(\begin{matrix}10&-342&399\\-92&138&-161\\1116&93&186\end{matrix}\right))$$

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.

$$10det(\left(\begin{matrix}138&-161\\93&186\end{matrix}\right))-\left(-342det(\left(\begin{matrix}-92&-161\\1116&186\end{matrix}\right))\right)+399det(\left(\begin{matrix}-92&138\\1116&93\end{matrix}\right))$$

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

$$10\left(138\times 186-93\left(-161\right)\right)-\left(-342\left(-92\times 186-1116\left(-161\right)\right)\right)+399\left(-92\times 93-1116\times 138\right)$$

Simplify.

$$10\times 40641-\left(-342\times 162564\right)+399\left(-162564\right)$$

Add the terms to obtain the final result.

$$-8859738$$

Factor

$$-8859738$$