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det(\left(\begin{matrix}2&-1&6\\3&2&-13\\1&3&1\end{matrix}\right))
Find the determinant of the matrix using the method of diagonals.
\left(\begin{matrix}2&-1&6&2&-1\\3&2&-13&3&2\\1&3&1&1&3\end{matrix}\right)
Extend the original matrix by repeating the first two columns as the fourth and fifth columns.
2\times 2-\left(-13\right)+6\times 3\times 3=71
Starting at the upper left entry, multiply down along the diagonals, and add the resulting products.
2\times 6+3\left(-13\right)\times 2+3\left(-1\right)=-69
Starting at the lower left entry, multiply up along the diagonals, and add the resulting products.
71-\left(-69\right)
Subtract the sum of the upward diagonal products from the sum of the downward diagonal products.
140
Subtract -69 from 71.
det(\left(\begin{matrix}2&-1&6\\3&2&-13\\1&3&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}2&-13\\3&1\end{matrix}\right))-\left(-det(\left(\begin{matrix}3&-13\\1&1\end{matrix}\right))\right)+6det(\left(\begin{matrix}3&2\\1&3\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(2-3\left(-13\right)\right)-\left(-\left(3-\left(-13\right)\right)\right)+6\left(3\times 3-2\right)
For the 2\times 2 matrix \left(\begin{matrix}a&b\\c&d\end{matrix}\right), the determinant is ad-bc.
2\times 41-\left(-16\right)+6\times 7
Simplify.
140
Add the terms to obtain the final result.