Solve |1|{ccc}{2}&{1}&{0}{1}&{-1}&{-4}{-1}&{3}&{8}=

$$| 1 | \left| \begin{array} { c c c } { 2 } & { 1 } & { 0 } \\ { 1 } & { - 1 } & { - 4 } \\ { - 1 } & { 3 } & { 8 } \end{array} \right| =$$

Evaluate

$$4$$

Steps Using Sarrus' Rule

The absolute value of a real number a is a when a\geq 0, or -a when a<0. The absolute value of 1 is 1.

$$1det(\left(\begin{matrix}2&1&0\\1&-1&-4\\-1&3&8\end{matrix}\right))$$

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

$$det(\left(\begin{matrix}2&1&0\\1&-1&-4\\-1&3&8\end{matrix}\right))$$

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

$$\left(\begin{matrix}2&1&0&2&1\\1&-1&-4&1&-1\\-1&3&8&-1&3\end{matrix}\right)$$

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

$$2\left(-1\right)\times 8-4\left(-1\right)=-12$$

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

$$3\left(-4\right)\times 2+8=-16$$

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

$$-12-\left(-16\right)$$

Subtract -16 from -12.

$$4$$

Steps Using Expansion by Minors

The absolute value of a real number a is a when a\geq 0, or -a when a<0. The absolute value of 1 is 1.

$$1det(\left(\begin{matrix}2&1&0\\1&-1&-4\\-1&3&8\end{matrix}\right))$$

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

$$det(\left(\begin{matrix}2&1&0\\1&-1&-4\\-1&3&8\end{matrix}\right))$$

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.

$$2det(\left(\begin{matrix}-1&-4\\3&8\end{matrix}\right))-det(\left(\begin{matrix}1&-4\\-1&8\end{matrix}\right))$$

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

$$2\left(-8-3\left(-4\right)\right)-\left(8-\left(-\left(-4\right)\right)\right)$$

Simplify.

$$2\times 4-4$$

Add the terms to obtain the final result.

$$4$$

Factor

$$2^{2}$$