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det(\left(\begin{matrix}1&2&3\\2&1&1\\0&2&1\end{matrix}\right))
Find the determinant of the matrix using the method of diagonals.
\left(\begin{matrix}1&2&3&1&2\\2&1&1&2&1\\0&2&1&0&2\end{matrix}\right)
Extend the original matrix by repeating the first two columns as the fourth and fifth columns.
1+3\times 2\times 2=13
Starting at the upper left entry, multiply down along the diagonals, and add the resulting products.
2+2\times 2=6
Starting at the lower left entry, multiply up along the diagonals, and add the resulting products.
13-6
Subtract the sum of the upward diagonal products from the sum of the downward diagonal products.
7
Subtract 6 from 13.
det(\left(\begin{matrix}1&2&3\\2&1&1\\0&2&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&1\\2&1\end{matrix}\right))-2det(\left(\begin{matrix}2&1\\0&1\end{matrix}\right))+3det(\left(\begin{matrix}2&1\\0&2\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.
1-2-2\times 2+3\times 2\times 2
For the 2\times 2 matrix \left(\begin{matrix}a&b\\c&d\end{matrix}\right), the determinant is ad-bc.
-1-2\times 2+3\times 4
Simplify.
7
Add the terms to obtain the final result.