# identical solution spaces based on matrices of the same order and the same rank

I came across a statement that:

if A and B are matrices of the same order, and the rank of B = the rank of AB, then the solution space of the homogenous system $$AB \underline{x}=0$$ is identical to the solution space to the homogenous system $$B \underline{x}=0$$.

No proof is provided, and I am a bit puzzled at why this would be the case? Can anyone provide some insight?

Thank you!

The rows of $$AB$$ are linear combinations of the rows of $$B$$. If the ranks of $$B,AB$$ are identical, the rowspace of $$B$$ and $$AB$$ is the same, and the nullspace of $$B,AB$$ is also the same, being the orthogonal complement of the row space.
In general, $$x$$ will be a solution to $$ABx = 0$$ if either $$Bx = 0$$, or $$Bx \neq 0$$ and $$ABx = 0$$. Consequently, every solution to $$Bx = 0$$ is also a solution to $$ABx = 0$$.
If we are given than $$AB$$ and $$B$$ have the same rank, however, then we can conclude (by the rank-nullity theorem) that the nullspaces of $$AB$$ and $$B$$ have the same dimension. So, the solution space to $$Bx = 0$$ is a subspace of $$ABx$$, but both subspaces have the same dimension. So, the subspaces (i.e. the solutions spaces) must be identical.
• I'm not sure the first line is correct. Did you mean $ABx=0$? – Shubham Johri Jan 8 at 19:31