Det - Maple Help

Det

inert determinant

 Calling Sequence Det(A)

Parameters

 A - Matrix

Description

 • The Det function is a placeholder for representing the determinant of the matrix A.  It is used in conjunction with mod and modp1 which define the coefficient domain as described below.
 • The call $\mathrm{Det}\left(A\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}m$ computes the determinant of the matrix $A\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}m$ in characteristic m which may not not be prime.  The entries in A may be integers, rationals, polynomials, or in general, rational functions in parameters over a finite field.
 • The call $\mathrm{modp1}\left(\mathrm{Det}\left(A\right),p\right)$ computes the determinant of the matrix $A\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}p$ where p is a prime integer and the entries of A are modp1 polynomials using fraction-free Gaussian elimination.

Examples

 > $A≔\mathrm{Matrix}\left(\left[\left[2,3,1\right],\left[3,2,3\right],\left[0,3,2\right]\right]\right)$
 ${A}{≔}\left[\begin{array}{rrr}{2}& {3}& {1}\\ {3}& {2}& {3}\\ {0}& {3}& {2}\end{array}\right]$ (1)
 > $\mathrm{Det}\left(A\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}3$
 ${2}$ (2)
 > $\mathrm{Det}\left(A\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}6$
 ${5}$ (3)
 > $C≔\mathrm{Matrix}\left(\left[\left[x-2,3,1\right],\left[3,x-2,3\right],\left[0,3,x-2\right]\right]\right)$
 ${C}{≔}\left[\begin{array}{ccc}{x}{-}{2}& {3}& {1}\\ {3}& {x}{-}{2}& {3}\\ {0}& {3}& {x}{-}{2}\end{array}\right]$ (4)
 > $\mathrm{Det}\left(C\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}3$
 ${{x}}^{{3}}{+}{1}$ (5)
 > $\mathrm{Charpoly}\left(A,x\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}3$
 ${{x}}^{{3}}{+}{1}$ (6)
 > $\mathrm{alias}\left(\mathrm{α}=\mathrm{RootOf}\left({x}^{4}+x+1\right)\right):$
 > $A≔\mathrm{Matrix}\left(\left[\left[1,\mathrm{α},{\mathrm{α}}^{2}\right],\left[\mathrm{α},1,\mathrm{α}\right],\left[{\mathrm{α}}^{2},\mathrm{α},1\right]\right]\right)$
 ${A}{≔}\left[\begin{array}{ccc}{1}& {\mathrm{α}}& {{\mathrm{α}}}^{{2}}\\ {\mathrm{α}}& {1}& {\mathrm{α}}\\ {{\mathrm{α}}}^{{2}}& {\mathrm{α}}& {1}\end{array}\right]$ (7)
 > $\mathrm{Det}\left(A\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}2$
 ${\mathrm{α}}$ (8)
 > $A≔\mathrm{Matrix}\left(\left[\left[1-\mathrm{α},\frac{\mathrm{α}}{t},1-\mathrm{α}t\right],\left[1+\mathrm{α},\mathrm{α}t,1+\mathrm{α}t\right],\left[\mathrm{α},1-\frac{\mathrm{α}}{t},\mathrm{α}t\right]\right]\right)$
 ${A}{≔}\left[\begin{array}{ccc}{1}{-}{\mathrm{α}}& \frac{{\mathrm{α}}}{{t}}& {-}{\mathrm{α}}{}{t}{+}{1}\\ {1}{+}{\mathrm{α}}& {\mathrm{α}}{}{t}& {\mathrm{α}}{}{t}{+}{1}\\ {\mathrm{α}}& {1}{-}\frac{{\mathrm{α}}}{{t}}& {\mathrm{α}}{}{t}\end{array}\right]$ (9)
 > $\mathrm{collect}\left(\mathrm{Det}\left(A\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}2,t\right)$
 ${{\mathrm{α}}}^{{2}}{}{{t}}^{{2}}{+}{{\mathrm{α}}}^{{2}}{}{t}{+}{{\mathrm{α}}}^{{2}}{+}\frac{{{\mathrm{α}}}^{{2}}}{{t}}$ (10)
 >