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Integrate w.r.t. a
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det(\left(\begin{matrix}a&b&c\\1&-3&2\\2&0&-8\end{matrix}\right))
Find the determinant of the matrix using the method of diagonals.
\left(\begin{matrix}a&b&c&a&b\\1&-3&2&1&-3\\2&0&-8&2&0\end{matrix}\right)
Extend the original matrix by repeating the first two columns as the fourth and fifth columns.
a\left(-3\right)\left(-8\right)+b\times 2\times 2=24a+4b
Starting at the upper left entry, multiply down along the diagonals, and add the resulting products.
2\left(-3\right)c-8b=-8b-6c
Starting at the lower left entry, multiply up along the diagonals, and add the resulting products.
24a+4b-\left(-8b-6c\right)
Subtract the sum of the upward diagonal products from the sum of the downward diagonal products.
24a+12b+6c
Subtract -6c-8b from 24a+4b.
det(\left(\begin{matrix}a&b&c\\1&-3&2\\2&0&-8\end{matrix}\right))
Find the determinant of the matrix using the method of expansion by minors (also known as expansion by cofactors).
adet(\left(\begin{matrix}-3&2\\0&-8\end{matrix}\right))-bdet(\left(\begin{matrix}1&2\\2&-8\end{matrix}\right))+cdet(\left(\begin{matrix}1&-3\\2&0\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.
a\left(-3\right)\left(-8\right)-b\left(-8-2\times 2\right)+c\left(-2\left(-3\right)\right)
For the 2\times 2 matrix \left(\begin{matrix}a&b\\c&d\end{matrix}\right), the determinant is ad-bc.
a\times 24-b\left(-12\right)+c\times 6
Simplify.
24a+12b+6c
Add the terms to obtain the final result.