Solve {lll}{103}&{1}&{204}{199}&{2}&{345}{301}&{3}&{600}

$$\left| \begin{array} { l l l } { 103 } & { 1 } & { 204 } \\ { 199 } & { 2 } & { 345 } \\ { 301 } & { 3 } & { 600 } \end{array} \right|$$

Evaluate

$$420$$

Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}103&1&204\\199&2&345\\301&3&600\end{matrix}\right))$$

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

$$\left(\begin{matrix}103&1&204&103&1\\199&2&345&199&2\\301&3&600&301&3\end{matrix}\right)$$

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

$$103\times 2\times 600+345\times 301+204\times 199\times 3=349233$$

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

$$301\times 2\times 204+3\times 345\times 103+600\times 199=348813$$

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

$$349233-348813$$

Subtract 348813 from 349233.

$$420$$

Steps Using Expansion by Minors

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

$$det(\left(\begin{matrix}103&1&204\\199&2&345\\301&3&600\end{matrix}\right))$$

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.

$$103det(\left(\begin{matrix}2&345\\3&600\end{matrix}\right))-det(\left(\begin{matrix}199&345\\301&600\end{matrix}\right))+204det(\left(\begin{matrix}199&2\\301&3\end{matrix}\right))$$

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

$$103\left(2\times 600-3\times 345\right)-\left(199\times 600-301\times 345\right)+204\left(199\times 3-301\times 2\right)$$

Simplify.

$$103\times 165-15555+204\left(-5\right)$$

Add the terms to obtain the final result.

$$420$$

Factor

$$2^{2}\times 3\times 5\times 7$$