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det(\left(\begin{matrix}-1&2&0\\1&3&4\\1&3&2\end{matrix}\right))
Find the determinant of the matrix using the method of diagonals.
\left(\begin{matrix}-1&2&0&-1&2\\1&3&4&1&3\\1&3&2&1&3\end{matrix}\right)
Extend the original matrix by repeating the first two columns as the fourth and fifth columns.
-3\times 2+2\times 4=2
Starting at the upper left entry, multiply down along the diagonals, and add the resulting products.
3\times 4\left(-1\right)+2\times 2=-8
Starting at the lower left entry, multiply up along the diagonals, and add the resulting products.
2-\left(-8\right)
Subtract the sum of the upward diagonal products from the sum of the downward diagonal products.
10
Subtract -8 from 2.
det(\left(\begin{matrix}-1&2&0\\1&3&4\\1&3&2\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}3&4\\3&2\end{matrix}\right))-2det(\left(\begin{matrix}1&4\\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(3\times 2-3\times 4\right)-2\left(2-4\right)
For the 2\times 2 matrix \left(\begin{matrix}a&b\\c&d\end{matrix}\right), the determinant is ad-bc.
-\left(-6\right)-2\left(-2\right)
Simplify.
10
Add the terms to obtain the final result.