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Calculate Determinant
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det(\left(\begin{matrix}2&-2&-5\\1&6&-3\\2&1&-1\end{matrix}\right))
Find the determinant of the matrix using the method of diagonals.
\left(\begin{matrix}2&-2&-5&2&-2\\1&6&-3&1&6\\2&1&-1&2&1\end{matrix}\right)
Extend the original matrix by repeating the first two columns as the fourth and fifth columns.
2\times 6\left(-1\right)-2\left(-3\right)\times 2-5=-5
Starting at the upper left entry, multiply down along the diagonals, and add the resulting products.
2\times 6\left(-5\right)-3\times 2-\left(-2\right)=-64
Starting at the lower left entry, multiply up along the diagonals, and add the resulting products.
-5-\left(-64\right)
Subtract the sum of the upward diagonal products from the sum of the downward diagonal products.
59
Subtract -64 from -5.
det(\left(\begin{matrix}2&-2&-5\\1&6&-3\\2&1&-1\end{matrix}\right))
Find the determinant of the matrix using the method of expansion by minors (also known as expansion by cofactors).
2det(\left(\begin{matrix}6&-3\\1&-1\end{matrix}\right))-\left(-2det(\left(\begin{matrix}1&-3\\2&-1\end{matrix}\right))\right)-5det(\left(\begin{matrix}1&6\\2&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.
2\left(6\left(-1\right)-\left(-3\right)\right)-\left(-2\left(-1-2\left(-3\right)\right)\right)-5\left(1-2\times 6\right)
For the 2\times 2 matrix \left(\begin{matrix}a&b\\c&d\end{matrix}\right), the determinant is ad-bc.
2\left(-3\right)-\left(-2\times 5\right)-5\left(-11\right)
Simplify.
59
Add the terms to obtain the final result.