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Calculate Determinant
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det(\left(\begin{matrix}2&1&-1\\2&1&0\\1&-1&1\end{matrix}\right))
Find the determinant of the matrix using the method of diagonals.
\left(\begin{matrix}2&1&-1&2&1\\2&1&0&2&1\\1&-1&1&1&-1\end{matrix}\right)
Extend the original matrix by repeating the first two columns as the fourth and fifth columns.
2-2\left(-1\right)=4
Starting at the upper left entry, multiply down along the diagonals, and add the resulting products.
-1+2=1
Starting at the lower left entry, multiply up along the diagonals, and add the resulting products.
4-1
Subtract the sum of the upward diagonal products from the sum of the downward diagonal products.
3
Subtract 1 from 4.
det(\left(\begin{matrix}2&1&-1\\2&1&0\\1&-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}1&0\\-1&1\end{matrix}\right))-det(\left(\begin{matrix}2&0\\1&1\end{matrix}\right))-det(\left(\begin{matrix}2&1\\1&-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.
2-2-\left(2\left(-1\right)-1\right)
For the 2\times 2 matrix \left(\begin{matrix}a&b\\c&d\end{matrix}\right), the determinant is ad-bc.
2-2-\left(-3\right)
Simplify.
3
Add the terms to obtain the final result.