Let a be a full rank matrix, B and C be n * t matrices. It is proved that the sufficient and necessary condition for ab = BC is b = C
Is ab = AC
Necessity: because AB = AC
So a (B-C) = 0
So the column vectors of B-C are the solutions of the homogeneous linear equations AX = 0
But a column with full rank, ax = 0, has only zero solution
So B-C = 0
So B = C
The sufficiency is obvious
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