Solve {ccc}{6}&{-2}&{9}{-2}&{3}&{-1}{9}&{-1}&{3}

Evaluate

Factor

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det(\left(\begin{matrix}6&-2&9\\-2&3&-1\\9&-1&3\end{matrix}\right))

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

\left(\begin{matrix}6&-2&9&6&-2\\-2&3&-1&-2&3\\9&-1&3&9&-1\end{matrix}\right)

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

6\times 3\times 3-2\left(-1\right)\times 9+9\left(-2\right)\left(-1\right)=90

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

9\times 3\times 9-\left(-6\right)+3\left(-2\right)\left(-2\right)=261

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

90-261

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

-171

Subtract 261 from 90.

det(\left(\begin{matrix}6&-2&9\\-2&3&-1\\9&-1&3\end{matrix}\right))

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

6det(\left(\begin{matrix}3&-1\\-1&3\end{matrix}\right))-\left(-2det(\left(\begin{matrix}-2&-1\\9&3\end{matrix}\right))\right)+9det(\left(\begin{matrix}-2&3\\9&-1\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.

6\left(3\times 3-\left(-\left(-1\right)\right)\right)-\left(-2\left(-2\times 3-9\left(-1\right)\right)\right)+9\left(-2\left(-1\right)-9\times 3\right)

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

6\times 8-\left(-2\times 3\right)+9\left(-25\right)

Simplify.

-171

Add the terms to obtain the final result.