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det(\left(\begin{matrix}1&4&7\\2&5&8\\3&6&9\end{matrix}\right))
Find the determinant of the matrix using the method of diagonals.
\left(\begin{matrix}1&4&7&1&4\\2&5&8&2&5\\3&6&9&3&6\end{matrix}\right)
Extend the original matrix by repeating the first two columns as the fourth and fifth columns.
5\times 9+4\times 8\times 3+7\times 2\times 6=225
Starting at the upper left entry, multiply down along the diagonals, and add the resulting products.
3\times 5\times 7+6\times 8+9\times 2\times 4=225
Starting at the lower left entry, multiply up along the diagonals, and add the resulting products.
225-225
Subtract the sum of the upward diagonal products from the sum of the downward diagonal products.
0
Subtract 225 from 225.
det(\left(\begin{matrix}1&4&7\\2&5&8\\3&6&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&8\\6&9\end{matrix}\right))-4det(\left(\begin{matrix}2&8\\3&9\end{matrix}\right))+7det(\left(\begin{matrix}2&5\\3&6\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.
5\times 9-6\times 8-4\left(2\times 9-3\times 8\right)+7\left(2\times 6-3\times 5\right)
For the 2\times 2 matrix \left(\begin{matrix}a&b\\c&d\end{matrix}\right), the determinant is ad-bc.
-3-4\left(-6\right)+7\left(-3\right)
Simplify.
0
Add the terms to obtain the final result.