Solve {lll}{1}&{2}&{3}{0}&{7}&{8}{1}&{1}&{2}

$$\left| \begin{array} { l l l } { 1 } & { 2 } & { 3 } \\ { 0 } & { 7 } & { 8 } \\ { 1 } & { 1 } & { 2 } \end{array} \right|$$

Evaluate

$$1$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}1&2&3\\0&7&8\\1&1&2\end{matrix}\right))$$

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

$$\left(\begin{matrix}1&2&3&1&2\\0&7&8&0&7\\1&1&2&1&1\end{matrix}\right)$$

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

$$7\times 2+2\times 8=30$$

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

$$7\times 3+8=29$$

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

$$30-29$$

Subtract 29 from 30.

$$1$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}1&2&3\\0&7&8\\1&1&2\end{matrix}\right))$$

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.

$$det(\left(\begin{matrix}7&8\\1&2\end{matrix}\right))-2det(\left(\begin{matrix}0&8\\1&2\end{matrix}\right))+3det(\left(\begin{matrix}0&7\\1&1\end{matrix}\right))$$

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

$$7\times 2-8-2\left(-8\right)+3\left(-7\right)$$

Simplify.

$$6-2\left(-8\right)+3\left(-7\right)$$

Add the terms to obtain the final result.

$$1$$

Factor

$$1$$