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det(\left(\begin{matrix}13&11&1\\5&17&0\\1&6&-2\end{matrix}\right))
Find the determinant of the matrix using the method of diagonals.
\left(\begin{matrix}13&11&1&13&11\\5&17&0&5&17\\1&6&-2&1&6\end{matrix}\right)
Extend the original matrix by repeating the first two columns as the fourth and fifth columns.
13\times 17\left(-2\right)+5\times 6=-412
Starting at the upper left entry, multiply down along the diagonals, and add the resulting products.
17-2\times 5\times 11=-93
Starting at the lower left entry, multiply up along the diagonals, and add the resulting products.
-412-\left(-93\right)
Subtract the sum of the upward diagonal products from the sum of the downward diagonal products.
-319
Subtract -93 from -412.
det(\left(\begin{matrix}13&11&1\\5&17&0\\1&6&-2\end{matrix}\right))
Find the determinant of the matrix using the method of expansion by minors (also known as expansion by cofactors).
13det(\left(\begin{matrix}17&0\\6&-2\end{matrix}\right))-11det(\left(\begin{matrix}5&0\\1&-2\end{matrix}\right))+det(\left(\begin{matrix}5&17\\1&6\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.
13\times 17\left(-2\right)-11\times 5\left(-2\right)+5\times 6-17
For the 2\times 2 matrix \left(\begin{matrix}a&b\\c&d\end{matrix}\right), the determinant is ad-bc.
13\left(-34\right)-11\left(-10\right)+13
Simplify.
-319
Add the terms to obtain the final result.