Solve {ccc}{265}&{240}&{219}{240}&{225}&{198}{219}&{198}&{181}

$$\left| \begin{array} { c c c } { 265 } & { 240 } & { 219 } \\ { 240 } & { 225 } & { 198 } \\ { 219 } & { 198 } & { 181 } \end{array} \right|$$

Evaluate

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Steps Using Sarrus' Rule

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

$$det(\left(\begin{matrix}265&240&219\\240&225&198\\219&198&181\end{matrix}\right))$$

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

$$\left(\begin{matrix}265&240&219&265&240\\240&225&198&240&225\\219&198&181&219&198\end{matrix}\right)$$

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

$$265\times 225\times 181+240\times 198\times 219+219\times 240\times 198=31605885$$

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

$$219\times 225\times 219+198\times 198\times 265+181\times 240\times 240=31605885$$

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

$$31605885-31605885$$

Subtract 31605885 from 31605885.

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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).

$$det(\left(\begin{matrix}265&240&219\\240&225&198\\219&198&181\end{matrix}\right))$$

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.

$$265det(\left(\begin{matrix}225&198\\198&181\end{matrix}\right))-240det(\left(\begin{matrix}240&198\\219&181\end{matrix}\right))+219det(\left(\begin{matrix}240&225\\219&198\end{matrix}\right))$$

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

$$265\left(225\times 181-198\times 198\right)-240\left(240\times 181-219\times 198\right)+219\left(240\times 198-219\times 225\right)$$

Simplify.

$$265\times 1521-240\times 78+219\left(-1755\right)$$

Add the terms to obtain the final result.

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Factor

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