Inverse - Maple Help

Inverse

inert matrix inverse

 Calling Sequence Inverse(A) mod n

Parameters

 A - Matrix n - integer, the modulus

Description

 • The Inverse function is a placeholder for representing the inverse of a square matrix A.
 • The call Inverse(A) mod n computes the inverse of the square matrix A over a finite ring of characteristic n. This includes finite fields, GF(p), the integers mod p, and GF(p^k) where elements of GF(p^k) are expressed as polynomials in RootOf's.

Examples

 > $A≔\mathrm{Matrix}\left(\left[\left[1,2,3\right],\left[1,3,0\right],\left[1,4,3\right]\right]\right)$
 ${A}{≔}\left[\begin{array}{ccc}{1}& {2}& {3}\\ {1}& {3}& {0}\\ {1}& {4}& {3}\end{array}\right]$ (1)
 > $B≔\mathrm{Inverse}\left(A\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}5$
 ${B}{≔}\left[\begin{array}{ccc}{4}& {1}& {1}\\ {2}& {0}& {3}\\ {1}& {3}& {1}\end{array}\right]$ (2)
 > $\mathrm{~}\left[\mathrm{Expand}\right]\left(A·B\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}5$
 $\left[\begin{array}{ccc}{1}& {0}& {0}\\ {0}& {1}& {0}\\ {0}& {0}& {1}\end{array}\right]$ (3)

The matrix A is singular mod 2:

 > $\mathrm{Inverse}\left(A\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}2$
 > $\mathrm{alias}\left(a=\mathrm{RootOf}\left({x}^{4}+x+1\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}2\right):$
 > $A≔\mathrm{Matrix}\left(\left[\left[1,a,{a}^{2}\right],\left[1,{a}^{2},1\right],\left[1,{a}^{3},{a}^{2}\right]\right]\right)$
 ${A}{≔}\left[\begin{array}{ccc}{1}& {a}& {{a}}^{{2}}\\ {1}& {{a}}^{{2}}& {1}\\ {1}& {{a}}^{{3}}& {{a}}^{{2}}\end{array}\right]$ (4)
 > $B≔\mathrm{Inverse}\left(A\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}2$
 ${B}{≔}\left[\begin{array}{ccc}{{a}}^{{2}}{+}{a}& {{a}}^{{3}}{+}{a}& {{a}}^{{3}}{+}{{a}}^{{2}}{+}{1}\\ {{a}}^{{3}}{+}{{a}}^{{2}}& {0}& {{a}}^{{3}}{+}{{a}}^{{2}}\\ {a}{+}{1}& {{a}}^{{3}}{+}{a}{+}{1}& {{a}}^{{3}}\end{array}\right]$ (5)
 > $\mathrm{~}\left[\mathrm{Expand}\right]\left(A·B\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}2$
 $\left[\begin{array}{ccc}{1}& {0}& {0}\\ {0}& {1}& {0}\\ {0}& {0}& {1}\end{array}\right]$ (6)