Solve {lll}{1}&{-3}&{3}{3}&{-5}&{3}{6}&{-6}&{4}

$$\left( \begin{array} { l l l } { 1 } & { - 3 } & { 3 } \\ { 3 } & { - 5 } & { 3 } \\ { 6 } & { - 6 } & { 4 } \end{array} \right)$$

Calculate Determinant

$$16$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}1&-3&3\\3&-5&3\\6&-6&4\end{matrix}\right))$$

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

$$\left(\begin{matrix}1&-3&3&1&-3\\3&-5&3&3&-5\\6&-6&4&6&-6\end{matrix}\right)$$

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

$$-5\times 4-3\times 3\times 6+3\times 3\left(-6\right)=-128$$

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

$$6\left(-5\right)\times 3-6\times 3+4\times 3\left(-3\right)=-144$$

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

$$-128-\left(-144\right)$$

Subtract -144 from -128.

$$16$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}1&-3&3\\3&-5&3\\6&-6&4\end{matrix}\right))$$

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.

$$det(\left(\begin{matrix}-5&3\\-6&4\end{matrix}\right))-\left(-3det(\left(\begin{matrix}3&3\\6&4\end{matrix}\right))\right)+3det(\left(\begin{matrix}3&-5\\6&-6\end{matrix}\right))$$

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

$$-5\times 4-\left(-6\times 3\right)-\left(-3\left(3\times 4-6\times 3\right)\right)+3\left(3\left(-6\right)-6\left(-5\right)\right)$$

Simplify.

$$-2-\left(-3\left(-6\right)\right)+3\times 12$$

Add the terms to obtain the final result.

$$16$$

Evaluate

$$\left(\begin{matrix}1&-3&3\\3&-5&3\\6&-6&4\end{matrix}\right)$$