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

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

Calculate Determinant

$$-33$$

Steps Using Sarrus' Rule

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

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

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

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

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

$$-4\times 2\left(-2\right)-3\left(-5\right)=31$$

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

$$2\times 2+3\left(-5\right)\left(-4\right)=64$$

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

$$31-64$$

Subtract 64 from 31.

$$-33$$

Steps Using Expansion by Minors

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

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

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.

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

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

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

Simplify.

$$2\left(-2\right)-\left(-4\left(-11\right)\right)-3\left(-5\right)$$

Add the terms to obtain the final result.

$$-33$$

Evaluate

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