The parameter A must be a square (n x n) Matrix of values from R.

The (indexed) parameter R, which specifies the domain of computation, a commutative ring, must be a Maple table/module which has the following values/exports:

R[`0`] : a constant for the zero of the ring R

R[`1`] : a constant for the (multiplicative) identity of R

R[`+`] : a procedure for adding elements of R (nary)

R[`-`] : a procedure for negating and subtracting elements of R (unary and binary)

R[`*`] : a procedure for multiplying elements of R (binary and commutative)

R[`=`] : a boolean procedure for testing if two elements of R are equal

The optional argument method=... specifies the algorithm to be used. The specific algorithms are as follows:

method=MinorExpansion directs the code to use minor expansion. This algorithm uses O(n 2^n) arithmetic operations in R.

method=BerkowitzAlgorithm directs the code to use the Berkowitz algorithm. This algorithm uses O(n^4) arithmetic operations in R.

method=BareissAlgorithm directs the code to use the Bareiss algorithm. This algorithm uses O(n^3) arithmetic operations in R but requires exact division, i.e., it requires R to be an integral domain with the following operation defined:

R[Divide]: a boolean procedure for dividing two elements of R where R[Divide](a,b,'q') outputs true if b | a and optionally assigns q the quotient such that a = b q.

method=GaussianElimination directs the code to use the Gaussian elimination algorithm. This algorithm uses O(n^3) arithmetic operations in R but requires R to be a field, i.e., the following operation must be defined:

R[`/`]: a procedure for dividing two elements of R

If the method is not given and the operation R[Divide] is defined, then the Bareiss algorithm is used, otherwise if the operation R[`/`] is defined then GaussianElimination is used, otherwise the Berkowitz algorithm is used.

$\mathrm{with}\left(\mathrm{LinearAlgebra}\left[\mathrm{Generic}\right]\right)\:$

$Z\left[\mathrm{`0`}\right],Z\left[\mathrm{`1`}\right],Z\left[\mathrm{`+`}\right],Z\left[\mathrm{`-`}\right],Z\left[\mathrm{`*`}\right],Z\left[\mathrm{`=`}\right]≔0,1,\mathrm{`+`},\mathrm{`-`},\mathrm{`*`},\mathrm{`=`}\:$

$A≔\mathrm{Matrix}\left(\left[\left[2\,1\,4\right]\,\left[3\,2\,1\right]\,\left[0\,0\,5\right]\right]\right)$

$A≔\left[\begin{array}{ccc}2& 1& 4\\ 3& 2& 1\\ 0& 0& 5\end{array}\right]$

$\mathrm{Determinant}\left[Z\right]\left(A\right)$

$5$

$Q\left[\mathrm{`0`}\right],Q\left[\mathrm{`1`}\right],Q\left[\mathrm{`+`}\right],Q\left[\mathrm{`-`}\right],Q\left[\mathrm{`*`}\right],Q\left[\mathrm{`/`}\right],Q\left[\mathrm{`=`}\right]≔0,1,\mathrm{`+`},\mathrm{`-`},\mathrm{`*`},\mathrm{`/`},\mathrm{`=`}\:$

$A≔\mathrm{Matrix}\left(\left[\left[2\,1\,4\,6\right]\,\left[3\,2\,1\,7\right]\,\left[0\,0\,5\,1\right]\,\left[0\,0\,3\,8\right]\right]\right)$

$A≔\left[\begin{array}{cccc}2& 1& 4& 6\\ 3& 2& 1& 7\\ 0& 0& 5& 1\\ 0& 0& 3& 8\end{array}\right]$

$\mathrm{Determinant}\left[Q\right]\left(A\right)$

$37$

$\mathrm{Determinant}\left[Q\right]\left(A\,\mathrm{method}=\mathrm{BerkowitzAlgorithm}\right)$

$37$