Solve {rrr}{0.5}&{-0.8}&{0.6}{0.2}&{0.9}&{0.7}{3.1}&{4.1}&{-2.8}

$$\left| \begin{array} { r r r } { 0.5 } & { - 0.8 } & { 0.6 } \\ { 0.2 } & { 0.9 } & { 0.7 } \\ { 3.1 } & { 4.1 } & { - 2.8 } \end{array} \right|$$

Evaluate

$$-6.061$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}0.5&-0.8&0.6\\0.2&0.9&0.7\\3.1&4.1&-2.8\end{matrix}\right))$$

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

$$\left(\begin{matrix}0.5&-0.8&0.6&0.5&-0.8\\0.2&0.9&0.7&0.2&0.9\\3.1&4.1&-2.8&3.1&4.1\end{matrix}\right)$$

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

$$0.5\times 0.9\left(-2.8\right)-0.8\times 0.7\times 3.1+0.6\times 0.2\times 4.1=-2.504$$

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

$$3.1\times 0.9\times 0.6+4.1\times 0.7\times 0.5-2.8\times 0.2\left(-0.8\right)=3.557$$

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

$$-2.504-3.557$$

Subtract 3.557 from -2.504 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-6.061$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}0.5&-0.8&0.6\\0.2&0.9&0.7\\3.1&4.1&-2.8\end{matrix}\right))$$

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.

$$0.5det(\left(\begin{matrix}0.9&0.7\\4.1&-2.8\end{matrix}\right))-\left(-0.8det(\left(\begin{matrix}0.2&0.7\\3.1&-2.8\end{matrix}\right))\right)+0.6det(\left(\begin{matrix}0.2&0.9\\3.1&4.1\end{matrix}\right))$$

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

$$0.5\left(0.9\left(-2.8\right)-4.1\times 0.7\right)-\left(-0.8\left(0.2\left(-2.8\right)-3.1\times 0.7\right)\right)+0.6\left(0.2\times 4.1-3.1\times 0.9\right)$$

Simplify.

$$0.5\left(-5.39\right)-\left(-0.8\left(-2.73\right)\right)+0.6\left(-1.97\right)$$

Add the terms to obtain the final result.

$$-6.061$$

Factor

$$-6.061$$