Solve {lll}{1}&{2}&{3}{1}&{3}&{5}{3}&{4}&{3}

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

Evaluate

$$-2$$

Steps Using Sarrus' Rule

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

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

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

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

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

$$3\times 3+2\times 5\times 3+3\times 4=51$$

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

$$3\times 3\times 3+4\times 5+3\times 2=53$$

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

$$51-53$$

Subtract 53 from 51.

$$-2$$

Steps Using Expansion by Minors

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

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

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.

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

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

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

Simplify.

$$-11-2\left(-12\right)+3\left(-5\right)$$

Add the terms to obtain the final result.

$$-2$$

Factor

$$-2$$