Solve {lll}{1}&{2}&{3}{1}&{1}&{0}{9}&{9}&{10}

$$\left| \begin{array} { l l l } { 1 } & { 2 } & { 3 } \\ { 1 } & { 1 } & { 0 } \\ { 9 } & { 9 } & { 10 } \end{array} \right|$$

Evaluate

$$-10$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}1&2&3\\1&1&0\\9&9&10\end{matrix}\right))$$

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

$$\left(\begin{matrix}1&2&3&1&2\\1&1&0&1&1\\9&9&10&9&9\end{matrix}\right)$$

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

$$10+3\times 9=37$$

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

$$9\times 3+10\times 2=47$$

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

$$37-47$$

Subtract 47 from 37.

$$-10$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}1&2&3\\1&1&0\\9&9&10\end{matrix}\right))$$

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.

$$det(\left(\begin{matrix}1&0\\9&10\end{matrix}\right))-2det(\left(\begin{matrix}1&0\\9&10\end{matrix}\right))+3det(\left(\begin{matrix}1&1\\9&9\end{matrix}\right))$$

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

$$10-2\times 10+3\left(9-9\right)$$

Simplify.

$$10-2\times 10$$

Add the terms to obtain the final result.

$$-10$$

Factor

$$-10$$