Solve Matrix

$$[ \begin{array} { c c c } { 3 } & { 4 } & { 5 } \\ { 2 } & { \frac { 1 } { 2 } } & { 4 } \\ { - 3 } & { - 1 } & { - 2 } \end{array} )$$

Calculate Determinant

$$-\frac{51}{2} = -25\frac{1}{2} = -25.5$$

Steps Using Sarrus' Rule

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

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

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

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

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

$$3\times \frac{1}{2}\left(-2\right)+4\times 4\left(-3\right)+5\times 2\left(-1\right)=-61$$

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

$$-3\times \frac{1}{2}\times 5-4\times 3-2\times 2\times 4=-\frac{71}{2}$$

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

$$-61-\left(-\frac{71}{2}\right)$$

Subtract -\frac{71}{2} from -61.

$$-\frac{51}{2}$$

Steps Using Expansion by Minors

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

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

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.

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

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

$$3\left(\frac{1}{2}\left(-2\right)-\left(-4\right)\right)-4\left(2\left(-2\right)-\left(-3\times 4\right)\right)+5\left(2\left(-1\right)-\left(-3\times \frac{1}{2}\right)\right)$$

Simplify.

$$3\times 3-4\times 8+5\left(-\frac{1}{2}\right)$$

Add the terms to obtain the final result.

$$-\frac{51}{2}$$

Evaluate

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