Solve {ccc}{3}&{-2}&{4}{2}&{-4}&{5}{1}&{8}&{2}

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

Calculate Determinant

$$-66$$

Steps Using Sarrus' Rule

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

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

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

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

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

$$3\left(-4\right)\times 2-2\times 5+4\times 2\times 8=30$$

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

$$-4\times 4+8\times 5\times 3+2\times 2\left(-2\right)=96$$

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

$$30-96$$

Subtract 96 from 30.

$$-66$$

Steps Using Expansion by Minors

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

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

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.

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

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

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

Simplify.

$$3\left(-48\right)-\left(-2\left(-1\right)\right)+4\times 20$$

Add the terms to obtain the final result.

$$-66$$

Evaluate

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