Solve {ccc}{1}&{2}&{2}{1}&{-5}&{-6}{3}&{-1}&{-2}

$$\left( \begin{array} { c c c } { 1 } & { 2 } & { 2 } \\ { 1 } & { - 5 } & { - 6 } \\ { 3 } & { - 1 } & { - 2 } \end{array} \right)$$

Calculate Determinant

$$0$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}1&2&2\\1&-5&-6\\3&-1&-2\end{matrix}\right))$$

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

$$\left(\begin{matrix}1&2&2&1&2\\1&-5&-6&1&-5\\3&-1&-2&3&-1\end{matrix}\right)$$

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

$$-5\left(-2\right)+2\left(-6\right)\times 3+2\left(-1\right)=-28$$

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

$$3\left(-5\right)\times 2-\left(-6\right)-2\times 2=-28$$

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

$$-28-\left(-28\right)$$

Subtract -28 from -28.

$$0$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}1&2&2\\1&-5&-6\\3&-1&-2\end{matrix}\right))$$

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.

$$det(\left(\begin{matrix}-5&-6\\-1&-2\end{matrix}\right))-2det(\left(\begin{matrix}1&-6\\3&-2\end{matrix}\right))+2det(\left(\begin{matrix}1&-5\\3&-1\end{matrix}\right))$$

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

$$-5\left(-2\right)-\left(-\left(-6\right)\right)-2\left(-2-3\left(-6\right)\right)+2\left(-1-3\left(-5\right)\right)$$

Simplify.

$$4-2\times 16+2\times 14$$

Add the terms to obtain the final result.

$$0$$

Evaluate

$$\left(\begin{matrix}1&2&2\\1&-5&-6\\3&-1&-2\end{matrix}\right)$$