The MatrixInverse(A) function, where A is a nonsingular square Matrix, returns the Matrix inverse $A^{\mathrm{-1}}$. If A is recognized as a singular Matrix, an error message is returned.

If A is a nonsingular $nxn$ Matrix, the inverse $A^{\mathrm{-1}}$ is computed such that $A·A^{\mathrm{-1}}=\mathrm{I}$, where I is the $nxn$ identity Matrix.

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

$M≔⟨⟨a\,c⟩|⟨b\,d⟩⟩$

$M≔\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

$\frac{1}{ad-bc}⟨⟨d\,-c⟩|⟨-b\,a⟩⟩$

$\left[\begin{array}{cc}\frac{d}{ad-bc}& -\frac{b}{ad-bc}\\ -\frac{c}{ad-bc}& \frac{a}{ad-bc}\end{array}\right]$

$\mathrm{M\_I}≔\mathrm{MatrixInverse}\left(M\right)$

$\mathrm{M\_I}≔\left[\begin{array}{cc}\frac{d}{ad-bc}& -\frac{b}{ad-bc}\\ -\frac{c}{ad-bc}& \frac{a}{ad-bc}\end{array}\right]$

$\mathrm{simplify}\left(M·\mathrm{M\_I}\right)$

$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

$P≔⟨⟨1\,2⟩\left|⟨3\,4⟩\right|⟨1\,7⟩⟩$

$P≔\left[\begin{array}{ccc}1& 3& 1\\ 2& 4& 7\end{array}\right]$

$\mathrm{Pseudoinverse}\left(P\right)$

$\left[\begin{array}{cc}\frac{9}{106}& \frac{1}{318}\\ \frac{41}{106}& -\frac{19}{318}\\ -\frac{13}{53}& \frac{28}{159}\end{array}\right]$

$P·\mathrm{Pseudoinverse}\left(P\right)$

$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

The Student[LinearAlgebra][MatrixInverse] command was introduced in Maple 2020.

For more information on Maple 2020 changes, see Updates in Maple 2020.