inverse of a non-square matrix returns a matrix instead of erroring

This issue was triaged from bugs/dan/0-inverse-matrix, one of the 857 files removed from the pre-GitHub bugs/ tree by d2c8d27826 and catalogued in #36. The commentary below was written by Claude (Claude Opus 5, via Claude Code), not by @d-torrance, whose account posted it -- please weigh it accordingly.

The original file, verbatim

all of these should give errors:

    o34 = new HashTable from {asdf => foo}

    i35 : inverse matrix {{1,2}}

    o35 = | 1 |
	  | 0 |

		   2        1
    o35 : Matrix ZZ  <--- ZZ

    i36 : ( matrix {{1,2}} )^-1

    o36 = | 1 |
	  | 0 |

		   2        1
    o36 : Matrix ZZ  <--- ZZ

    i39 : ( matrix {{1},{2}} )^-1
    stdio:39:21:(3):[0]: error: matrix not invertible

    i40 : inverse matrix {{1},{2}}
    stdio:40:1:(3):[0]: error: matrix not invertible

Where it stands today

Still reproduces. A non-square matrix gets an "inverse" instead of an error:

i1 : inverse matrix {{1,2}}
o1 = | 1 |
     | 0 |
                2        1
o1 : Matrix ZZ  <--- ZZ

i2 : (matrix {{1,2}})^-1              -- same answer by the other spelling

while the transpose is diagnosed correctly:

i3 : inverse matrix {{1},{2}}
stdio:3:1:(3): error: matrix not invertible

So one orientation errors and the other returns a matrix that is not an inverse of anything —
matrix {{1,2}} * matrix {{1},{0}} is the 1×1 identity, but the product the other way is not the 2×2
identity, which is what "inverse" claims.

Why the two orientations differ

The computation is a solve rather than an inversion: for a wide matrix a right inverse exists and is
found, for a tall one nothing satisfies the system and the error fires. The bug is that inverse does
not first insist on a square matrix, so a one-sided solution is returned under a name that promises a
two-sided one.

Related

#3738 reaches a wrong inverse by a different path, so the two are worth reading together but are
not the same defect.

_{open · disposition issue · source of truth: bug-triage/catalog.tsv}

Reproduce the two cases using the inverse matrix and ^-1 entry points described here, and compare them with the transpose case. Trace the inverse implementation and the historical bugs/dan/0-inverse-matrix report, then add regression coverage showing that non-square matrices error in both orientations while square inverses still work.