Solve {rrr}{1}&{2}&{-1}{0}&{2}&{3}{4}&{1}&{5}

$$\left| \begin{array} { r r r } { 1 } & { 2 } & { - 1 } \\ { 0 } & { 2 } & { 3 } \\ { 4 } & { 1 } & { 5 } \end{array} \right|$$

Evaluate

$$39$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}1&2&-1\\0&2&3\\4&1&5\end{matrix}\right))$$

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

$$\left(\begin{matrix}1&2&-1&1&2\\0&2&3&0&2\\4&1&5&4&1\end{matrix}\right)$$

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

$$2\times 5+2\times 3\times 4=34$$

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

$$4\times 2\left(-1\right)+3=-5$$

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

$$34-\left(-5\right)$$

Subtract -5 from 34.

$$39$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}1&2&-1\\0&2&3\\4&1&5\end{matrix}\right))$$

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.

$$det(\left(\begin{matrix}2&3\\1&5\end{matrix}\right))-2det(\left(\begin{matrix}0&3\\4&5\end{matrix}\right))-det(\left(\begin{matrix}0&2\\4&1\end{matrix}\right))$$

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

$$2\times 5-3-2\left(-4\times 3\right)-\left(-4\times 2\right)$$

Simplify.

$$7-2\left(-12\right)-\left(-8\right)$$

Add the terms to obtain the final result.

$$39$$

Factor

$$3\times 13$$