Are there any generalized inverses that would produce a left inverse for a short rectangular matrix?

To give some context, I’m trying to solve the following problem:

y=BA−1xy = BA^{-1}x

where:

yy = n×1n \times 1 vector — is known

xx = 3×13 \times 1 vector — is unknown

BB = 3×n3 \times n matrix — is known

AA = n×nn \times n singular matrix — So AA is known and cannot be inverted to solve the problem.

If I could compute a generalized inverse for BB (let’s call it B†B^\dagger) than I could avoid dealing with A−1A^{-1} altogether. The solution would simply be:

x=AB†yx = AB^\dagger y

However for this to work B†B^\dagger would have to be a left inverse, i.e. B†B=IB^\dagger B = I.

Any ideas on a generalized inverse method that would produce a left inverse for a short (less rows than columns) matrix? Is this even possible?

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What do you mean by A−1A^{-1} is AA is not invertible? And the size of your vectors and matrices are inconsistent with the product you write (maybe the sizes of xx and yy are inverted?), unless n=3n=3….
– Arnaud D.
yesterday

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