# 16.14 Inverse

## Description

The minverse X-Function generates an inverse matrix by dividing the adjoint by its determinant. When matrices do not have inverses or determinants, a Moore-Penrose pseudoinverse is computed.

1. Create a new matrix with data.
2. Activate the matrix.
3. Select Analysis: Mathematics: Inverse to open the minverse dialog box.

## Dialog Options

Recalculate Controls recalculation of analysis results None Auto Manual For more information, see: Recalculating Analysis Results Choose input matrix. For help with range controls, see: Specifying Your Input Data Choose where to output the inverse matrix. For help with the range controls, see: Output Results

## Algorithm

For a square matrix of full rank$A\!$, the inverse matrix $A^{-1}\!$ (aka the reciprocal matrix), will satisfy the relationship:

$AA\!^{-1}=A\!^{-1}\!A=I\!$

where $I\!$ is the identity matrix.

The calculation of $A^{-1}\!$ can be expressed as:

$A^{-1}=\frac 1{|A|}A^{*}$

where $|A|\!$ is the determinant of matrix $A\!$ and $A^*\!$is the adjoint.

$A^*=\begin{bmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{n1} & \cdots & a_{nn} \end{bmatrix}$

$a_{ij}=\left( -1\right) ^{i+j}|A^{ij}|$

where $A^{ij}\!$ is the $(n-1)\times (n-1)$ matrix by elimination of the $i_{th}\!$column and $j_{th}\!$row of $A\!$.

When matrices do not have inverses or determinants, a Moore-Penrose pseudoinverse will be computed. It exists for any $(m,n)\!$ matrix.

Given an $(m\times n)$ matrix $A\!$, $A^+\!$is the unique $n\times m\!$ pseudoinverse matrix. If $m>n\!$ and A has full rank, then the $A^+\!$ satisfies the following:

$A\!^{+}=(A^TA)\!^{-1}A\!^T$

The computation is based on singular value decomposition (SVD) of the matrix $A\!$ and any singular values within tolerance, are treated as zero. If the rank of $A\!$ is not full, then the matrix will shrink to a smaller matrix.

## References

1. E. H. Moore: On the reciprocal of the general algebraic matrix. Bulletin of the American Mathematical Society 26, 394-395 (1920).
2. Roger Penrose: A generalized inverse for matrices. Proceedings of the Cambridge Philosophical Society 51, 406-413 (1955).