Solve {ccc}{2}&{-3}&{6}{7}&{6}&{8}{-17}&{-24}&{-12}

$$\left| \begin{array} { c c c } { 2 } & { - 3 } & { 6 } \\ { 7 } & { 6 } & { 8 } \\ { - 17 } & { - 24 } & { - 12 } \end{array} \right|$$

Evaluate

$$0$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}2&-3&6\\7&6&8\\-17&-24&-12\end{matrix}\right))$$

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

$$\left(\begin{matrix}2&-3&6&2&-3\\7&6&8&7&6\\-17&-24&-12&-17&-24\end{matrix}\right)$$

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

$$2\times 6\left(-12\right)-3\times 8\left(-17\right)+6\times 7\left(-24\right)=-744$$

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

$$-17\times 6\times 6-24\times 8\times 2-12\times 7\left(-3\right)=-744$$

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

$$-744-\left(-744\right)$$

Subtract -744 from -744.

$$0$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}2&-3&6\\7&6&8\\-17&-24&-12\end{matrix}\right))$$

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.

$$2det(\left(\begin{matrix}6&8\\-24&-12\end{matrix}\right))-\left(-3det(\left(\begin{matrix}7&8\\-17&-12\end{matrix}\right))\right)+6det(\left(\begin{matrix}7&6\\-17&-24\end{matrix}\right))$$

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

$$2\left(6\left(-12\right)-\left(-24\times 8\right)\right)-\left(-3\left(7\left(-12\right)-\left(-17\times 8\right)\right)\right)+6\left(7\left(-24\right)-\left(-17\times 6\right)\right)$$

Simplify.

$$2\times 120-\left(-3\times 52\right)+6\left(-66\right)$$

Add the terms to obtain the final result.

$$0$$

Factor

$$0$$