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 parameters $A,B$, and $v,w$ must have compatible dimensions for the addition. Similarly, all Matrices in $\mathrm{lA}$, and all Vectors in $\mathrm{lv}$, must have compatible dimensions, and the number of elements of $\mathrm{lc}$, if given, must match the number of element of $\mathrm{lA}$ or $\mathrm{lv}$, respectively.

If $c,d,\mathrm{lc}$ are omitted, then MatrixAdd and VectorAdd just compute the sum of the matrices or vectors, respectively.

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

$\mathrm{F8}≔\mathrm{GF}\left(2\,3\,a^3+a+1\right)$

$\mathrm{F8}≔{𝔽}_8$

$A≔\mathrm{map}\left(\mathrm{F8}\,\mathrm{Matrix}\left(\left[\left[a\,a+1\,0\right]\,\left[a^2\,0\,a^2+a\right]\right]\right)\right)$

$A≔\left[\begin{array}{ccc}a\mathbf{mod}2& \left(a+1\right)\mathbf{mod}2& 0\mathbf{mod}2\\ a^2\mathbf{mod}2& 0\mathbf{mod}2& \left(a^2+a\right)\mathbf{mod}2\end{array}\right]$

$B≔\mathrm{map}\left(\mathrm{F8}\,\mathrm{Matrix}\left(\left[\left[1\,a^2+a\,a\right]\,\left[a^2+1\,a\,a\right]\right]\right)\right)$

$B≔\left[\begin{array}{ccc}1\mathbf{mod}2& \left(a^2+a\right)\mathbf{mod}2& a\mathbf{mod}2\\ \left(a^2+1\right)\mathbf{mod}2& a\mathbf{mod}2& a\mathbf{mod}2\end{array}\right]$

$\mathrm{MatrixAdd}\left[\mathrm{F8}\right]\left(A\,B\right)=\mathrm{MatrixAdd}\left[\mathrm{F8}\right]\left(\left[A\,B\right]\right)$

$\left[\begin{array}{ccc}\left(a+1\right)\mathbf{mod}2& \left(a^2+1\right)\mathbf{mod}2& a\mathbf{mod}2\\ 1\mathbf{mod}2& a\mathbf{mod}2& a^2\mathbf{mod}2\end{array}\right]=\left[\begin{array}{ccc}\left(a+1\right)\mathbf{mod}2& \left(a^2+1\right)\mathbf{mod}2& a\mathbf{mod}2\\ 1\mathbf{mod}2& a\mathbf{mod}2& a^2\mathbf{mod}2\end{array}\right]$

$c≔\mathrm{F8}\left(a\right)$

$c≔a\mathbf{mod}2$

$d≔\mathrm{F8}\left(a+1\right)$

$d≔\left(a+1\right)\mathbf{mod}2$

$\mathrm{MatrixAdd}\left[\mathrm{F8}\right]\left(A\,B\,c\,d\right)=\mathrm{MatrixAdd}\left[\mathrm{F8}\right]\left(\left[A\,B\right]\,\left[c\,d\right]\right)$

$\left[\begin{array}{ccc}\left(a^2+a+1\right)\mathbf{mod}2& \left(a^2+a+1\right)\mathbf{mod}2& \left(a^2+a\right)\mathbf{mod}2\\ \left(a^2+a+1\right)\mathbf{mod}2& \left(a^2+a\right)\mathbf{mod}2& 1\mathbf{mod}2\end{array}\right]=\left[\begin{array}{ccc}\left(a^2+a+1\right)\mathbf{mod}2& \left(a^2+a+1\right)\mathbf{mod}2& \left(a^2+a\right)\mathbf{mod}2\\ \left(a^2+a+1\right)\mathbf{mod}2& \left(a^2+a\right)\mathbf{mod}2& 1\mathbf{mod}2\end{array}\right]$

$v≔\mathrm{map}\left(\mathrm{F8}\,\mathrm{Vector}\left(\left[a^2+a+1\,a\,a+1\right]\right)\right)$

$v≔\left[\begin{array}{c}\left(a^2+a+1\right)\mathbf{mod}2\\ a\mathbf{mod}2\\ \left(a+1\right)\mathbf{mod}2\end{array}\right]$

$w≔\mathrm{map}\left(\mathrm{F8}\,\mathrm{Vector}\left(\left[a+1\,a^2\,a^2+a\right]\right)\right)$

$w≔\left[\begin{array}{c}\left(a+1\right)\mathbf{mod}2\\ a^2\mathbf{mod}2\\ \left(a^2+a\right)\mathbf{mod}2\end{array}\right]$

$\mathrm{VectorAdd}\left[\mathrm{F8}\right]\left(v\,w\right)=\mathrm{VectorAdd}\left[\mathrm{F8}\right]\left(\left[v\,w\right]\right)$

$\left[\begin{array}{c}{a}^2\mathbf{mod}2\\ \left(a^2+a\right)\mathbf{mod}2\\ \left(a^2+1\right)\mathbf{mod}2\end{array}\right]=\left[\begin{array}{c}{a}^2\mathbf{mod}2\\ \left(a^2+a\right)\mathbf{mod}2\\ \left(a^2+1\right)\mathbf{mod}2\end{array}\right]$

$\mathrm{VectorAdd}\left[\mathrm{F8}\right]\left(v\,w\,c\,d\right)=\mathrm{VectorAdd}\left[\mathrm{F8}\right]\left(\left[v\,w\right]\,\left[c\,d\right]\right)$

$\left[\begin{array}{c}0\mathbf{mod}2\\ \left(a+1\right)\mathbf{mod}2\\ \left(a^2+a+1\right)\mathbf{mod}2\end{array}\right]=\left[\begin{array}{c}0\mathbf{mod}2\\ \left(a+1\right)\mathbf{mod}2\\ \left(a^2+a+1\right)\mathbf{mod}2\end{array}\right]$

The LinearAlgebra[Generic][MatrixAdd] and LinearAlgebra[Generic][VectorAdd] commands were introduced in Maple 2021.

For more information on Maple 2021 changes, see Updates in Maple 2021.