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Calculate Determinant
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det(\left(\begin{matrix}-1&0&1\\2&5&3\\1&2&9\end{matrix}\right))
Find the determinant of the matrix using the method of diagonals.
\left(\begin{matrix}-1&0&1&-1&0\\2&5&3&2&5\\1&2&9&1&2\end{matrix}\right)
Extend the original matrix by repeating the first two columns as the fourth and fifth columns.
-5\times 9+2\times 2=-41
Starting at the upper left entry, multiply down along the diagonals, and add the resulting products.
5+2\times 3\left(-1\right)=-1
Starting at the lower left entry, multiply up along the diagonals, and add the resulting products.
-41-\left(-1\right)
Subtract the sum of the upward diagonal products from the sum of the downward diagonal products.
-40
Subtract -1 from -41.
det(\left(\begin{matrix}-1&0&1\\2&5&3\\1&2&9\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}5&3\\2&9\end{matrix}\right))+det(\left(\begin{matrix}2&5\\1&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.
-\left(5\times 9-2\times 3\right)+2\times 2-5
For the 2\times 2 matrix \left(\begin{matrix}a&b\\c&d\end{matrix}\right), the determinant is ad-bc.
-39-1
Simplify.
-40
Add the terms to obtain the final result.