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det(\left(\begin{matrix}1&-1&2\\-1&1&3\\3&0&-1\end{matrix}\right))
Find the determinant of the matrix using the method of diagonals.
\left(\begin{matrix}1&-1&2&1&-1\\-1&1&3&-1&1\\3&0&-1&3&0\end{matrix}\right)
Extend the original matrix by repeating the first two columns as the fourth and fifth columns.
-1-3\times 3=-10
Starting at the upper left entry, multiply down along the diagonals, and add the resulting products.
3\times 2-\left(-\left(-1\right)\right)=5
Starting at the lower left entry, multiply up along the diagonals, and add the resulting products.
-10-5
Subtract the sum of the upward diagonal products from the sum of the downward diagonal products.
-15
Subtract 5 from -10.
det(\left(\begin{matrix}1&-1&2\\-1&1&3\\3&0&-1\end{matrix}\right))
Find the determinant of the matrix using the method of expansion by minors (also known as expansion by cofactors).
det(\left(\begin{matrix}1&3\\0&-1\end{matrix}\right))-\left(-det(\left(\begin{matrix}-1&3\\3&-1\end{matrix}\right))\right)+2det(\left(\begin{matrix}-1&1\\3&0\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.
-1-\left(-\left(-\left(-1\right)-3\times 3\right)\right)+2\left(-3\right)
For the 2\times 2 matrix \left(\begin{matrix}a&b\\c&d\end{matrix}\right), the determinant is ad-bc.
-1-\left(-\left(-8\right)\right)+2\left(-3\right)
Simplify.
-15
Add the terms to obtain the final result.