Solve {ccc}{-3}&{2}&{1}{203}&{298}&{399}{1/3}&{1/2}&{2/3}

$$\left| \begin{array} { c c c } { - 3 } & { 2 } & { 1 } \\ { 203 } & { 298 } & { 399 } \\ { \frac { 1 } { 3 } } & { \frac { 1 } { 2 } } & { \frac { 2 } { 3 } } \end{array} \right|$$

Evaluate

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Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}-3&2&1\\203&298&399\\\frac{1}{3}&\frac{1}{2}&\frac{2}{3}\end{matrix}\right))$$

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

$$\left(\begin{matrix}-3&2&1&-3&2\\203&298&399&203&298\\\frac{1}{3}&\frac{1}{2}&\frac{2}{3}&\frac{1}{3}&\frac{1}{2}\end{matrix}\right)$$

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

$$-3\times 298\times \frac{2}{3}+2\times 399\times \frac{1}{3}+203\times \frac{1}{2}=-\frac{457}{2}$$

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

$$\frac{1}{3}\times 298+\frac{1}{2}\times 399\left(-3\right)+\frac{2}{3}\times 203\times 2=-\frac{457}{2}$$

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

$$-\frac{457}{2}-\left(-\frac{457}{2}\right)$$

Subtract -\frac{457}{2} from -\frac{457}{2} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

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Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}-3&2&1\\203&298&399\\\frac{1}{3}&\frac{1}{2}&\frac{2}{3}\end{matrix}\right))$$

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.

$$-3det(\left(\begin{matrix}298&399\\\frac{1}{2}&\frac{2}{3}\end{matrix}\right))-2det(\left(\begin{matrix}203&399\\\frac{1}{3}&\frac{2}{3}\end{matrix}\right))+det(\left(\begin{matrix}203&298\\\frac{1}{3}&\frac{1}{2}\end{matrix}\right))$$

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

$$-3\left(298\times \frac{2}{3}-\frac{1}{2}\times 399\right)-2\left(203\times \frac{2}{3}-\frac{1}{3}\times 399\right)+203\times \frac{1}{2}-\frac{1}{3}\times 298$$

Simplify.

$$-3\left(-\frac{5}{6}\right)-2\times \frac{7}{3}+\frac{13}{6}$$

Add the terms to obtain the final result.

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Factor

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