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If the above matrix is full rank, the [[Moore–Penrose inverse]] matrices of it and its transpose are
:<math>\begin{align}
\begin{bmatrix}\mathbf A & \mathbf B\end{bmatrix}^{+} &=
\left(
\begin{bmatrix}\mathbf A & \mathbf B\end{bmatrix}^T
\begin{bmatrix}\mathbf A & \mathbf B\end{bmatrix}
▲ \begin{bmatrix}\mathbf A & \mathbf B\end{bmatrix}^T,
\begin{bmatrix}
\mathbf A^T \\
\mathbf B^T
\end{bmatrix}^{+} &=
\begin{bmatrix}\mathbf A & \mathbf B\end{bmatrix} \left(
\begin{bmatrix}\mathbf A & \mathbf B\end{bmatrix}^T
\begin{bmatrix}\mathbf A & \mathbf B\end{bmatrix}
\right)^{-1}.
\end{align}</math>
This computation of the pseudoinverse requires (''n'' + ''p'')-square matrix inversion and does not take advantage of the block form.
To reduce computational costs to ''n''- and ''p''-square matrix inversions and to introduce parallelism, treating the blocks separately, one derives <ref name=Baksalary>{{cite journal|author=[[Jerzy Baksalary|J.K. Baksalary]] and O.M. Baksalary|title=Particular formulae for the Moore–Penrose inverse of a columnwise partitioned matrix|journal=Linear Algebra Appl.|volume=421|date=2007|pages=16–23|doi=10.1016/j.laa.2006.03.031}}</ref>
:<math>
\begin{bmatrix}\mathbf A & \mathbf B\end{bmatrix}^{+} &=
\begin{bmatrix}
\mathbf P_B^\perp \mathbf A\left(\mathbf A^T \mathbf P_B^\perp \mathbf A\right)^{-1} \\
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\left(\mathbf P_B^{\perp}\mathbf A\right)^{+} \\
\left(\mathbf P_A^{\perp}\mathbf B\right)^{+}
\end{bmatrix}, \\
\begin{bmatrix}
\mathbf A^T \\
\mathbf B^T
\end{bmatrix}^{+} &=
\begin{bmatrix}
\mathbf P_B^\perp \mathbf A\left(\mathbf A^T \mathbf P_B^\perp \mathbf A\right)^{-1},\quad
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\left(\mathbf B^T \mathbf P_A^{\perp}\right)^{+}
\end{bmatrix},
\end{align}</math>
where [[orthogonal projection]] matrices are defined by
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