Can Every Square Matrix be written as product of two commuting matrices. The title explains it all. Can every square matrix $A$ be written as $A=B_1B_2=B_2B_1$ of any two matrices $B_1$,$B_2$. 
 A: The answer is yes, and you can produce arbitrarily many such factorisations. Take any polynomial $P$ whose zeros are not eigenvalues of $A$, then $P(A)$ is invertible, so you can write $A=AP(A)\cdot P(A)^{-1}$. More generally, you can replace $P$ by an entire function (or power series that converges on a disk with radius bigger than the norm of $A$).
I think you have to add more details to your question, e.g. require $B_1$ and $B_2$ to satisfy some additional conditions.
Here's another question: Can you characterise all possible factorisations?
A: To answer a variant of the question in a comment by OP under the question: a diagonal matrix with distinct entries cannot be written as a product of two commuting non-diagonal matrices (in fact both commuting matrices need to be diagonal). If $A=BC=CB$ then $B$ commutes with $BC=A$ (similarly for $C$), and any matrix that commutes with $A$ must stabilise each of the eigenspaces of $A$; since here these are $1$-dimensional, this means $B$ and $C$ are diagonal matrices.
A: If the matrix is low rank (rank(A) < n) you can factorize the matrix using row-reduced echelon form of A. That produce A = CR. But A is not equal RC. So low rank matrix A can't be written as A = B1B2 = B2B1
