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det(\left(\begin{matrix}1&2&-2\\0&-5&2\\3&-4&0\end{matrix}\right))
Find the determinant of the matrix using the method of diagonals.
\left(\begin{matrix}1&2&-2&1&2\\0&-5&2&0&-5\\3&-4&0&3&-4\end{matrix}\right)
Extend the original matrix by repeating the first two columns as the fourth and fifth columns.
2\times 2\times 3=12
Starting at the upper left entry, multiply down along the diagonals, and add the resulting products.
3\left(-5\right)\left(-2\right)-4\times 2=22
Starting at the lower left entry, multiply up along the diagonals, and add the resulting products.
12-22
Subtract the sum of the upward diagonal products from the sum of the downward diagonal products.
-10
Subtract 22 from 12.
det(\left(\begin{matrix}1&2&-2\\0&-5&2\\3&-4&0\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&2\\-4&0\end{matrix}\right))-2det(\left(\begin{matrix}0&2\\3&0\end{matrix}\right))-2det(\left(\begin{matrix}0&-5\\3&-4\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(-4\times 2\right)-2\left(-3\times 2\right)-2\left(-3\left(-5\right)\right)
For the 2\times 2 matrix \left(\begin{matrix}a&b\\c&d\end{matrix}\right), the determinant is ad-bc.
8-2\left(-6\right)-2\times 15
Simplify.
-10
Add the terms to obtain the final result.