Is it possible to contruct a 4x3 matrix b and a 3x4 matrix c such that bc is invertible?

Matrix B has dimensions 4x3

Matrix C has dimensions 3x4

Write out those dimensions like so: 4x3 3x4

The inner '3's match up so B*C is possible. The dimensions of B*C is 4x4 as these are the outer dimensions when lining up 4x3 3x4

Since B*C is a square matrix, it is possible that an inverse could exist. Keep in mind that this isn't a guarantee as the determinant of B*C could be zero (which would lead to BC being non-invertible). Though of course we'd need more info.