Solve {ccc}{3}&{3}&{-4}{-4}&{3}&{3}{1}&{-1}&{-1}

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

Calculate Determinant

$$-7$$

Steps Using Sarrus' Rule

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

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

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

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

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

$$3\times 3\left(-1\right)+3\times 3-4\left(-4\right)\left(-1\right)=-16$$

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

$$3\left(-4\right)-3\times 3-\left(-4\times 3\right)=-9$$

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

$$-16-\left(-9\right)$$

Subtract -9 from -16.

$$-7$$

Steps Using Expansion by Minors

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

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

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.

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

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

$$3\left(3\left(-1\right)-\left(-3\right)\right)-3\left(-4\left(-1\right)-3\right)-4\left(-4\left(-1\right)-3\right)$$

Simplify.

$$-3-4$$

Add the terms to obtain the final result.

$$-7$$

Evaluate

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