Solve {rrr}{1}&{-2}&{3}{-3}&{4}&{-5}{-1}&{-7}&{8}

$$\left( \begin{array} { r r r } { 1 } & { - 2 } & { 3 } \\ { - 3 } & { 4 } & { - 5 } \\ { - 1 } & { - 7 } & { 8 } \end{array} \right)$$

Calculate Determinant

$$14$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}1&-2&3\\-3&4&-5\\-1&-7&8\end{matrix}\right))$$

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

$$\left(\begin{matrix}1&-2&3&1&-2\\-3&4&-5&-3&4\\-1&-7&8&-1&-7\end{matrix}\right)$$

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

$$4\times 8-2\left(-5\right)\left(-1\right)+3\left(-3\right)\left(-7\right)=85$$

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

$$-4\times 3-7\left(-5\right)+8\left(-3\right)\left(-2\right)=71$$

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

$$85-71$$

Subtract 71 from 85.

$$14$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}1&-2&3\\-3&4&-5\\-1&-7&8\end{matrix}\right))$$

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.

$$det(\left(\begin{matrix}4&-5\\-7&8\end{matrix}\right))-\left(-2det(\left(\begin{matrix}-3&-5\\-1&8\end{matrix}\right))\right)+3det(\left(\begin{matrix}-3&4\\-1&-7\end{matrix}\right))$$

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

$$4\times 8-\left(-7\left(-5\right)\right)-\left(-2\left(-3\times 8-\left(-\left(-5\right)\right)\right)\right)+3\left(-3\left(-7\right)-\left(-4\right)\right)$$

Simplify.

$$-3-\left(-2\left(-29\right)\right)+3\times 25$$

Add the terms to obtain the final result.

$$14$$

Evaluate

$$\left(\begin{matrix}1&-2&3\\-3&4&-5\\-1&-7&8\end{matrix}\right)$$