\left| \begin{array} { l l l } { 10 ! } & { 11 ! } & { 12 ! } \\ { 11 ! } & { 12 ! } & { 13 ! } \\ { 12 ! } & { 13 ! } & { 14 ! } \end{array} \right|
Evaluate
138766843838988288000000
Factor
2^{27}\times 3^{13}\times 5^{6}\times 7^{3}\times 11^{2}
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det(\left(\begin{matrix}3628800&11!&12!\\11!&12!&13!\\12!&13!&14!\end{matrix}\right))
The factorial of 10 is 3628800.
det(\left(\begin{matrix}3628800&39916800&12!\\11!&12!&13!\\12!&13!&14!\end{matrix}\right))
The factorial of 11 is 39916800.
det(\left(\begin{matrix}3628800&39916800&479001600\\11!&12!&13!\\12!&13!&14!\end{matrix}\right))
The factorial of 12 is 479001600.
det(\left(\begin{matrix}3628800&39916800&479001600\\39916800&12!&13!\\12!&13!&14!\end{matrix}\right))
The factorial of 11 is 39916800.
det(\left(\begin{matrix}3628800&39916800&479001600\\39916800&479001600&13!\\12!&13!&14!\end{matrix}\right))
The factorial of 12 is 479001600.
det(\left(\begin{matrix}3628800&39916800&479001600\\39916800&479001600&6227020800\\12!&13!&14!\end{matrix}\right))
The factorial of 13 is 6227020800.
det(\left(\begin{matrix}3628800&39916800&479001600\\39916800&479001600&6227020800\\479001600&13!&14!\end{matrix}\right))
The factorial of 12 is 479001600.
det(\left(\begin{matrix}3628800&39916800&479001600\\39916800&479001600&6227020800\\479001600&6227020800&14!\end{matrix}\right))
The factorial of 13 is 6227020800.
det(\left(\begin{matrix}3628800&39916800&479001600\\39916800&479001600&6227020800\\479001600&6227020800&87178291200\end{matrix}\right))
The factorial of 14 is 87178291200.
\left(\begin{matrix}3628800&39916800&479001600&3628800&39916800\\39916800&479001600&6227020800&39916800&479001600\\479001600&6227020800&87178291200&479001600&6227020800\end{matrix}\right)
Extend the original matrix by repeating the first two columns as the fourth and fifth columns.
3628800\times 479001600\times 87178291200+39916800\times 6227020800\times 479001600+479001600\times 39916800\times 6227020800=389657297499879112704000000
Starting at the upper left entry, multiply down along the diagonals, and add the resulting products.
479001600\times 479001600\times 479001600+6227020800\times 6227020800\times 3628800+87178291200\times 39916800\times 39916800=389518530656040124416000000
Starting at the lower left entry, multiply up along the diagonals, and add the resulting products.
389657297499879112704000000-389518530656040124416000000
Subtract the sum of the upward diagonal products from the sum of the downward diagonal products.
138766843838988288000000
Subtract 389518530656040124416000000 from 389657297499879112704000000.
det(\left(\begin{matrix}3628800&11!&12!\\11!&12!&13!\\12!&13!&14!\end{matrix}\right))
The factorial of 10 is 3628800.
det(\left(\begin{matrix}3628800&39916800&12!\\11!&12!&13!\\12!&13!&14!\end{matrix}\right))
The factorial of 11 is 39916800.
det(\left(\begin{matrix}3628800&39916800&479001600\\11!&12!&13!\\12!&13!&14!\end{matrix}\right))
The factorial of 12 is 479001600.
det(\left(\begin{matrix}3628800&39916800&479001600\\39916800&12!&13!\\12!&13!&14!\end{matrix}\right))
The factorial of 11 is 39916800.
det(\left(\begin{matrix}3628800&39916800&479001600\\39916800&479001600&13!\\12!&13!&14!\end{matrix}\right))
The factorial of 12 is 479001600.
det(\left(\begin{matrix}3628800&39916800&479001600\\39916800&479001600&6227020800\\12!&13!&14!\end{matrix}\right))
The factorial of 13 is 6227020800.
det(\left(\begin{matrix}3628800&39916800&479001600\\39916800&479001600&6227020800\\479001600&13!&14!\end{matrix}\right))
The factorial of 12 is 479001600.
det(\left(\begin{matrix}3628800&39916800&479001600\\39916800&479001600&6227020800\\479001600&6227020800&14!\end{matrix}\right))
The factorial of 13 is 6227020800.
det(\left(\begin{matrix}3628800&39916800&479001600\\39916800&479001600&6227020800\\479001600&6227020800&87178291200\end{matrix}\right))
The factorial of 14 is 87178291200.
3628800det(\left(\begin{matrix}479001600&6227020800\\6227020800&87178291200\end{matrix}\right))-39916800det(\left(\begin{matrix}39916800&6227020800\\479001600&87178291200\end{matrix}\right))+479001600det(\left(\begin{matrix}39916800&479001600\\479001600&6227020800\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.
3628800\left(479001600\times 87178291200-6227020800\times 6227020800\right)-39916800\left(39916800\times 87178291200-479001600\times 6227020800\right)+479001600\left(39916800\times 6227020800-479001600\times 479001600\right)
For the 2\times 2 matrix \left(\begin{matrix}a&b\\c&d\end{matrix}\right), the determinant is ad-bc.
3628800\times 2982752926433280000-39916800\times 497125487738880000+479001600\times 19120211066880000
Simplify.
138766843838988288000000
Add the terms to obtain the final result.
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