Solve {ccc}{-7}&{-1}&{1}{-6}&{0}&{1/2}{-1}&{1}&{1}

$$\left| \begin{array} { c c c } { - 7 } & { - 1 } & { 1 } \\ { - 6 } & { 0 } & { \frac { 1 } { 2 } } \\ { - 1 } & { 1 } & { 1 } \end{array} \right|$$

Evaluate

$$-8$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}-7&-1&1\\-6&0&\frac{1}{2}\\-1&1&1\end{matrix}\right))$$

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

$$\left(\begin{matrix}-7&-1&1&-7&-1\\-6&0&\frac{1}{2}&-6&0\\-1&1&1&-1&1\end{matrix}\right)$$

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

$$-\frac{1}{2}\left(-1\right)-6=-\frac{11}{2}$$

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

$$\frac{1}{2}\left(-7\right)-6\left(-1\right)=\frac{5}{2}$$

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

$$-\frac{11}{2}-\frac{5}{2}$$

Subtract \frac{5}{2} from -\frac{11}{2} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-8$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}-7&-1&1\\-6&0&\frac{1}{2}\\-1&1&1\end{matrix}\right))$$

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.

$$-7det(\left(\begin{matrix}0&\frac{1}{2}\\1&1\end{matrix}\right))-\left(-det(\left(\begin{matrix}-6&\frac{1}{2}\\-1&1\end{matrix}\right))\right)+det(\left(\begin{matrix}-6&0\\-1&1\end{matrix}\right))$$

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

$$-7\left(-\frac{1}{2}\right)-\left(-\left(-6-\left(-\frac{1}{2}\right)\right)\right)-6$$

Simplify.

$$-7\left(-\frac{1}{2}\right)-\left(-\left(-\frac{11}{2}\right)\right)-6$$

Add the terms to obtain the final result.

$$-8$$

Factor

$$-8$$