# `⌹` — Matrix inverse / Matrix divide

Keys: `Alt-q u`. Ranks: `2` monadic, `∞ 2` dyadic

`⌹Y` gives an inverse, or a full-column-rank least-squares
pseudoinverse. Units reciprocate.

``` bpl
⌹2ₓ                ⍝ 1r2
⌹[1 0⋄0 2]ₓ     ⍝ [1ₓ 0ₓ⋄0ₓ 1r2]
```

On an array of rank 3 or more, `⌹Y` inverts each matrix, and `X⌹Y`
divides `X` by each matrix.

``` bpl
⌹2 2 2⍴[1 0 0 1 2 0 0 4]ₓ   ⍝ 2 2 2⍴[1 0 0 1 1r2 0 0 1r4]ₓ
```

`X⌹Y` solves `Y+.×Z = X`, using least squares when overdetermined.

``` bpl
[3 5 7]ₓ⌹[1 1⋄1 2⋄1 3]ₓ      ⍝ [1 2]ₓ
```

All-exact inputs use rational elimination. Approximate real/complex
inputs use pivoted LU for square systems and SVD for rectangular least
squares.

## Inverse

`⌹⁻¹Y` is `⌹Y`.

``` bpl
⌹⁻¹[1 0⋄0 2]ₓ   ⍝ [1 0⋄0 1r2]ₓ
```

## Errors

- `DOMAIN`: singular or underdetermined system, non-finite input