Solve {lll}{1}&{2}&{1}{3}&{4}&{2}{1}&{2}&{2}

$$\left( \begin{array} { l l l } { 1 } & { 2 } & { 1 } \\ { 3 } & { 4 } & { 2 } \\ { 1 } & { 2 } & { 2 } \end{array} \right)$$

Calculate Determinant

$$-2$$

Steps Using Sarrus' Rule

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

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

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

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

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

$$4\times 2+2\times 2+3\times 2=18$$

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

$$4+2\times 2+2\times 3\times 2=20$$

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

$$18-20$$

Subtract 20 from 18.

$$-2$$

Steps Using Expansion by Minors

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

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

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.

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

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

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

Simplify.

$$4-2\times 4+2$$

Add the terms to obtain the final result.

$$-2$$

Evaluate

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