The Adjoint(A) function constructs Matrix M such that $A·M=\mathrm{Determinant}\left(A\right)·\mathrm{IdentityMatrix}\left(\mathrm{Dimensions}\left(A\right)\right)$.  This is known as the "adjugate", "adjunct" or classical adjoint of A.

The constructor options provide additional information (readonly, shape, storage, order, datatype, and attributes) to the Matrix constructor that builds the result. These options may also be provided in the form outputoptions=[...], where [...] represents a Maple list.  If a constructor option is provided in both the calling sequence directly and in an outputoptions option, the latter takes precedence (regardless of the order).

This function is part of the LinearAlgebra package, and so it can be used in the form Adjoint(..) only after executing the command with(LinearAlgebra). However, it can always be accessed through the long form of the command by using LinearAlgebra[Adjoint](..).

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

$\mathrm{A1}≔⟨⟨9\,4\,1⟩\left|⟨1\,3\,-1⟩\right|⟨0\,8\,1⟩⟩\:$

$\mathrm{C1}≔\mathrm{Adjoint}\left(\mathrm{A1}\,\mathrm{datatype}=\mathrm{float}\right)$

$\mathrm{C1}≔\left[\begin{array}{ccc}11.& \mathrm{-1.}& 8.\\ 4.& 9.& \mathrm{-72.}\\ \mathrm{-7.}& 10.& 23.\end{array}\right]$

$\mathrm{A1}·\mathrm{C1}$

$\left[\begin{array}{ccc}103.& 0.& 0.\\ 0.& 103.& 0.\\ 0.& 0.& 103.\end{array}\right]$

$\mathrm{Determinant}\left(\mathrm{A1}\right)$

$103$

$\mathrm{A2}≔⟨⟨a\,2a⟩|⟨3\,-a⟩⟩\:$

$\mathrm{C2}≔\mathrm{Adjoint}\left(\mathrm{A2}\right)$

$\mathrm{C2}≔\left[\begin{array}{cc}-a& \mathrm{-3}\\ -2a& a\end{array}\right]$

$\mathrm{A2}·\mathrm{C2}$

$\left[\begin{array}{cc}-a^2-6a& 0\\ 0& -a^2-6a\end{array}\right]$

$\mathrm{Determinant}\left(\mathrm{A2}\right)$

$-a^2-6a$

de Boor, Carl. "An Empty Exercise." ACM SIGNUM Newsletter, Vol. 25 No. 2. (1990): 2-6.