Inverting non-square matrix for bootstrapping

Hi, attached is an example of a topic I am currently looking into, some basic application of linear algebra in something called yield curve construction.
Any mathematicians may excuse my inexact language here.
I acknowledge this may chiefly be more of a math question than a (pure) Excel one.

My example is simplified for the purpose of illustrating how this works and what my problem is.

I am given 4 bonds with a certain cash flow profile, i.e. mathematically speaking basically 4 polynoms of degrees one to four which are not dependant upon each other in any way.
In rows 27 to 43, instead of iteratively as shown in the previous rows in the spreadsheet, I solved the equations with inverting the coefficient matrix and multiplying it with the constants matrix.

So far, so good.
The numbers and the technique employed makes sense to me (first block coloured in gray and which reconciles to the previous manual solutions).

To my problem:
Now I was trying to construct the same result vector but with a slightly reduced set of equations to work with.
I was taking equations 1, 2 and 4 of the previous example, so left out Bonds #3 which gives me a non-square matrix.
Whether or not there's still a solution to this matrix, I guess depends among others on its determinant.
In order to be able to invert the matrix, I multiplied both sides with the transposed matrix of A and solved in the same way as before.

I am now a bit puzzled at the results of this, the second gray vector.
Especially, given that the first two equations are the same as before these two equations alone should be sufficient to solve x1 and x2 of the variable matrix X,
how can the solutions be different than before?


Maybe someone can shed some light on whether my matrix transformations are incorrect technically or mathematically or what I am missing about the interpretation of the results.

Thank and Regards