Det - Maple Programming 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 $\mathrm{Det}\left(A\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}m$ computes the determinant of the matrix $A\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}m$ 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 $A\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}p$ where p is a prime integer and the entries of A are modp1 polynomials using fraction-free Gaussian elimination.

Examples

 > A := Matrix([[2,3,1],[3,2,3],[0,3,2]]);
 ${A}{≔}\left[\begin{array}{ccc}{2}& {3}& {1}\\ {3}& {2}& {3}\\ {0}& {3}& {2}\end{array}\right]$ (1)
 > Det(A) mod 3;
 ${2}$ (2)
 > Det(A) mod 6;
 ${5}$ (3)
 > C := Matrix([[x-2,3,1],[3,x-2,3],[0,3,x-2]]);
 ${C}{≔}\left[\begin{array}{ccc}{x}{-}{2}& {3}& {1}\\ {3}& {x}{-}{2}& {3}\\ {0}& {3}& {x}{-}{2}\end{array}\right]$ (4)
 > Det(C) mod 3;
 ${{x}}^{{3}}{+}{1}$ (5)
 > Charpoly(A,x) mod 3;
 ${{x}}^{{3}}{+}{1}$ (6)
 > alias(alpha=RootOf(x^4+x+1)): # GF(16)
 > A := Matrix([[1,alpha,alpha^2],              [alpha,1,alpha],              [alpha^2,alpha,1]] );
 ${A}{≔}\left[\begin{array}{ccc}{1}& {\mathrm{\alpha }}& {{\mathrm{\alpha }}}^{{2}}\\ {\mathrm{\alpha }}& {1}& {\mathrm{\alpha }}\\ {{\mathrm{\alpha }}}^{{2}}& {\mathrm{\alpha }}& {1}\end{array}\right]$ (7)
 > Det(A) mod 2;
 ${\mathrm{\alpha }}$ (8)
 > A := Matrix([[1-alpha,alpha/t,1-alpha*t],              [1+alpha,alpha*t,1+alpha*t],              [alpha, 1-alpha/t, alpha*t]]) ;
 ${A}{≔}\left[\begin{array}{ccc}{1}{-}{\mathrm{\alpha }}& \frac{{\mathrm{\alpha }}}{{t}}& {-}{\mathrm{\alpha }}{}{t}{+}{1}\\ {1}{+}{\mathrm{\alpha }}& {\mathrm{\alpha }}{}{t}& {\mathrm{\alpha }}{}{t}{+}{1}\\ {\mathrm{\alpha }}& {1}{-}\frac{{\mathrm{\alpha }}}{{t}}& {\mathrm{\alpha }}{}{t}\end{array}\right]$ (9)
 > collect( Det(A) mod 2, t );
 ${{\mathrm{\alpha }}}^{{2}}{}{{t}}^{{2}}{+}{{\mathrm{\alpha }}}^{{2}}{}{t}{+}{{\mathrm{\alpha }}}^{{2}}{+}\frac{{{\mathrm{\alpha }}}^{{2}}}{{t}}$ (10)