Solve {rrr}{-10}&{-3}&{6}{8}&{4}&{2}{5}&{-7}&{9}

$$\left( \begin{array} { r r r } { - 10 } & { - 3 } & { 6 } \\ { 8 } & { 4 } & { 2 } \\ { 5 } & { - 7 } & { 9 } \end{array} \right)$$

Calculate Determinant

$$-770$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}-10&-3&6\\8&4&2\\5&-7&9\end{matrix}\right))$$

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

$$\left(\begin{matrix}-10&-3&6&-10&-3\\8&4&2&8&4\\5&-7&9&5&-7\end{matrix}\right)$$

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

$$-10\times 4\times 9-3\times 2\times 5+6\times 8\left(-7\right)=-726$$

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

$$5\times 4\times 6-7\times 2\left(-10\right)+9\times 8\left(-3\right)=44$$

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

$$-726-44$$

Subtract 44 from -726.

$$-770$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}-10&-3&6\\8&4&2\\5&-7&9\end{matrix}\right))$$

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.

$$-10det(\left(\begin{matrix}4&2\\-7&9\end{matrix}\right))-\left(-3det(\left(\begin{matrix}8&2\\5&9\end{matrix}\right))\right)+6det(\left(\begin{matrix}8&4\\5&-7\end{matrix}\right))$$

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

$$-10\left(4\times 9-\left(-7\times 2\right)\right)-\left(-3\left(8\times 9-5\times 2\right)\right)+6\left(8\left(-7\right)-5\times 4\right)$$

Simplify.

$$-10\times 50-\left(-3\times 62\right)+6\left(-76\right)$$

Add the terms to obtain the final result.

$$-770$$

Evaluate

$$\left(\begin{matrix}-10&-3&6\\8&4&2\\5&-7&9\end{matrix}\right)$$