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Calculate Determinant
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det(\left(\begin{matrix}1&2&3\\2&4&5\\3&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&4&5&2&4\\3&2&1&3&2\end{matrix}\right)
Extend the original matrix by repeating the first two columns as the fourth and fifth columns.
4+2\times 5\times 3+3\times 2\times 2=46
Starting at the upper left entry, multiply down along the diagonals, and add the resulting products.
3\times 4\times 3+2\times 5+2\times 2=50
Starting at the lower left entry, multiply up along the diagonals, and add the resulting products.
46-50
Subtract the sum of the upward diagonal products from the sum of the downward diagonal products.
-4
Subtract 50 from 46.
det(\left(\begin{matrix}1&2&3\\2&4&5\\3&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}4&5\\2&1\end{matrix}\right))-2det(\left(\begin{matrix}2&5\\3&1\end{matrix}\right))+3det(\left(\begin{matrix}2&4\\3&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.
4-2\times 5-2\left(2-3\times 5\right)+3\left(2\times 2-3\times 4\right)
For the 2\times 2 matrix \left(\begin{matrix}a&b\\c&d\end{matrix}\right), the determinant is ad-bc.
-6-2\left(-13\right)+3\left(-8\right)
Simplify.
-4
Add the terms to obtain the final result.