Solve {ccc}{3}&{2}&{6}{101}&{68}&{204}{257}&{170}&{512}=

$$\left| \begin{array} { c c c } { 3 } & { 2 } & { 6 } \\ { 101 } & { 68 } & { 204 } \\ { 257 } & { 170 } & { 512 } \end{array} \right| =$$

Evaluate

$$4$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}3&2&6\\101&68&204\\257&170&512\end{matrix}\right))$$

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

$$\left(\begin{matrix}3&2&6&3&2\\101&68&204&101&68\\257&170&512&257&170\end{matrix}\right)$$

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

$$3\times 68\times 512+2\times 204\times 257+6\times 101\times 170=312324$$

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

$$257\times 68\times 6+170\times 204\times 3+512\times 101\times 2=312320$$

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

$$312324-312320$$

Subtract 312320 from 312324.

$$4$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}3&2&6\\101&68&204\\257&170&512\end{matrix}\right))$$

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.

$$3det(\left(\begin{matrix}68&204\\170&512\end{matrix}\right))-2det(\left(\begin{matrix}101&204\\257&512\end{matrix}\right))+6det(\left(\begin{matrix}101&68\\257&170\end{matrix}\right))$$

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

$$3\left(68\times 512-170\times 204\right)-2\left(101\times 512-257\times 204\right)+6\left(101\times 170-257\times 68\right)$$

Simplify.

$$3\times 136-2\left(-716\right)+6\left(-306\right)$$

Add the terms to obtain the final result.

$$4$$

Factor

$$2^{2}$$