# [MATL], 19 bytes

 ,!ix"0GYs@12XQ!g]v!

Inputs are `M` (matrix), `H` (column vector), `V` (column vector). The row separator is `;`.

[Try it online!][TIO-j5ln4z6t] Or verify all test cases: [1](https://tio.run/##LYo9D8IgAAV3f8XRuQMPKLRhcXPupiEMTsaknTTGf48fYXnJ3b39@txaG839PdjT5XGUO6/mVl@mtSIcnsCUiSRmFmQzEnLIo4C@TRElNKMFZ@uhKPO7/bdzV/UD), [2](https://tio.run/##y00syfn/X0cxs0LJwD2y2MHQKCJQMT22TPH//2hDUwVDQzAyt1YwNFYwMlAwtFCwiOWKNrRWMAAjQygnFgA), [3](https://tio.run/##y00syfn/X0cxs0LJwD2y2MHQKCJQMT22TPH/f3MuQy5DAA), [4](https://tio.run/##y00syfn/X0cxs0LJwD2y2MHQKCJQMT22TPH//2hjawVdQ2sFk1guQ65oIAOIDGIB), [5](https://tio.run/##ZVA9SwUxEOz9FXnWErKbzeZCGjtrO@VxhZUIWikP//05O3fKA@Fy@zWZmc3Hy9f7tt2d3r5vy8Pz573o0@Ppdb2ctu0sJfXck@SB01LLhqjoLalkT5YraztmJdeZFM0KgGXhQEgRPUsOoobZiD5wPbeJQjDsKBT5ghxqIewoHbHinqJdUQfEJpUNuAApqIy0C5iUHoU2Oq148hCpQbqLO2I/0EI9ox0nXtKYSJ20BU4dx0Ioxy6N4oN2lXbjISAh4BzgCgd2LKihjjpebF9CdxsK@TGJb9ylUT0k65/7wXsRwzN3XG/OMlPhJ79/@Z9f467RZf0B).

[MATL]: https://github.com/lmendo/MATL
[TIO-j5ln4z6t]: https://tio.run/##LYo9D8IgAAV3f8XRuQMPKLRhcXPupiEMTsaknTTGf48fYXnJ3b39@txaG839PdjT5XGUO6/mVl@mtSIcnsCUiSRmFmQzEnLIo4C@TRElNKMFZ@uhKPO7/bdzV/UD "MATL – Try It Online"

_I'll add an explanation tomorrow._