Solve {ccc}{4}&{-10}&{2}{-10}&{14}&{26}{2}&{26}&{-34}

$$\left| \begin{array} { c c c } { 4 } & { - 10 } & { 2 } \\ { - 10 } & { 14 } & { 26 } \\ { 2 } & { 26 } & { - 34 } \end{array} \right|$$

Evaluate

$$-2304$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}4&-10&2\\-10&14&26\\2&26&-34\end{matrix}\right))$$

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

$$\left(\begin{matrix}4&-10&2&4&-10\\-10&14&26&-10&14\\2&26&-34&2&26\end{matrix}\right)$$

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

$$4\times 14\left(-34\right)-10\times 26\times 2+2\left(-10\right)\times 26=-2944$$

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

$$2\times 14\times 2+26\times 26\times 4-34\left(-10\right)\left(-10\right)=-640$$

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

$$-2944-\left(-640\right)$$

Subtract -640 from -2944.

$$-2304$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}4&-10&2\\-10&14&26\\2&26&-34\end{matrix}\right))$$

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.

$$4det(\left(\begin{matrix}14&26\\26&-34\end{matrix}\right))-\left(-10det(\left(\begin{matrix}-10&26\\2&-34\end{matrix}\right))\right)+2det(\left(\begin{matrix}-10&14\\2&26\end{matrix}\right))$$

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

$$4\left(14\left(-34\right)-26\times 26\right)-\left(-10\left(-10\left(-34\right)-2\times 26\right)\right)+2\left(-10\times 26-2\times 14\right)$$

Simplify.

$$4\left(-1152\right)-\left(-10\times 288\right)+2\left(-288\right)$$

Add the terms to obtain the final result.

$$-2304$$

Factor

$$-2304$$