Solve {lll}{33}&{10}&{13}{62}&{20}&{23}{93}&{30}&{33}

$$\left| \begin{array} { l l l } { 33 } & { 10 } & { 13 } \\ { 62 } & { 20 } & { 23 } \\ { 93 } & { 30 } & { 33 } \end{array} \right|$$

Evaluate

$$-60$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}33&10&13\\62&20&23\\93&30&33\end{matrix}\right))$$

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

$$\left(\begin{matrix}33&10&13&33&10\\62&20&23&62&20\\93&30&33&93&30\end{matrix}\right)$$

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

$$33\times 20\times 33+10\times 23\times 93+13\times 62\times 30=67350$$

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

$$93\times 20\times 13+30\times 23\times 33+33\times 62\times 10=67410$$

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

$$67350-67410$$

Subtract 67410 from 67350.

$$-60$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}33&10&13\\62&20&23\\93&30&33\end{matrix}\right))$$

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.

$$33det(\left(\begin{matrix}20&23\\30&33\end{matrix}\right))-10det(\left(\begin{matrix}62&23\\93&33\end{matrix}\right))+13det(\left(\begin{matrix}62&20\\93&30\end{matrix}\right))$$

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

$$33\left(20\times 33-30\times 23\right)-10\left(62\times 33-93\times 23\right)+13\left(62\times 30-93\times 20\right)$$

Simplify.

$$33\left(-30\right)-10\left(-93\right)$$

Add the terms to obtain the final result.

$$-60$$

Factor

$$-60$$