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Calculate Determinant
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det(\left(\begin{matrix}2&3&1\\1&5&0\\7&1&2\end{matrix}\right))
Find the determinant of the matrix using the method of diagonals.
\left(\begin{matrix}2&3&1&2&3\\1&5&0&1&5\\7&1&2&7&1\end{matrix}\right)
Extend the original matrix by repeating the first two columns as the fourth and fifth columns.
2\times 5\times 2+1=21
Starting at the upper left entry, multiply down along the diagonals, and add the resulting products.
7\times 5+2\times 3=41
Starting at the lower left entry, multiply up along the diagonals, and add the resulting products.
21-41
Subtract the sum of the upward diagonal products from the sum of the downward diagonal products.
-20
Subtract 41 from 21.
det(\left(\begin{matrix}2&3&1\\1&5&0\\7&1&2\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}5&0\\1&2\end{matrix}\right))-3det(\left(\begin{matrix}1&0\\7&2\end{matrix}\right))+det(\left(\begin{matrix}1&5\\7&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\times 5\times 2-3\times 2+1-7\times 5
For the 2\times 2 matrix \left(\begin{matrix}a&b\\c&d\end{matrix}\right), the determinant is ad-bc.
2\times 10-3\times 2-34
Simplify.
-20
Add the terms to obtain the final result.