Solve {ccc}{2}&{-1}&{9}{-7}&{-3}&{-4}{2}&{-1}&{-4}

$$\left( \begin{array} { c c c } { 2 } & { - 1 } & { 9 } \\ { - 7 } & { - 3 } & { - 4 } \\ { 2 } & { - 1 } & { - 4 } \end{array} \right)$$

Calculate Determinant

$$169$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}2&-1&9\\-7&-3&-4\\2&-1&-4\end{matrix}\right))$$

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

$$\left(\begin{matrix}2&-1&9&2&-1\\-7&-3&-4&-7&-3\\2&-1&-4&2&-1\end{matrix}\right)$$

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

$$2\left(-3\right)\left(-4\right)-\left(-4\times 2\right)+9\left(-7\right)\left(-1\right)=95$$

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

$$2\left(-3\right)\times 9-\left(-4\times 2\right)-4\left(-7\right)\left(-1\right)=-74$$

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

$$95-\left(-74\right)$$

Subtract -74 from 95.

$$169$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}2&-1&9\\-7&-3&-4\\2&-1&-4\end{matrix}\right))$$

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.

$$2det(\left(\begin{matrix}-3&-4\\-1&-4\end{matrix}\right))-\left(-det(\left(\begin{matrix}-7&-4\\2&-4\end{matrix}\right))\right)+9det(\left(\begin{matrix}-7&-3\\2&-1\end{matrix}\right))$$

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

$$2\left(-3\left(-4\right)-\left(-\left(-4\right)\right)\right)-\left(-\left(-7\left(-4\right)-2\left(-4\right)\right)\right)+9\left(-7\left(-1\right)-2\left(-3\right)\right)$$

Simplify.

$$2\times 8-\left(-36\right)+9\times 13$$

Add the terms to obtain the final result.

$$169$$

Evaluate

$$\left(\begin{matrix}2&-1&9\\-7&-3&-4\\2&-1&-4\end{matrix}\right)$$