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I have to prove something for my matrix-algebra course. it's the following proof:

I have to prove that $A\otimes B$ is invertible, if and only if $B\otimes B$ is invertible.

Please explain this in simple language, I'm only a first year econometrics student.

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Are $A$ and $B$ square matrices? – Pete L. Clark Feb 16 '13 at 17:27
Could be, it's not given.. – Sjoerd Smaal Feb 16 '13 at 17:29
This might help give you some intuition: – snarski Feb 16 '13 at 18:03
I think $B\otimes B$ should be corrected to $B\otimes A$. See,Invertibility of a Kronecker Product – M.Sina Feb 16 '13 at 18:43
up vote 2 down vote accepted

This is wrong. Take the zero matrix as $A$ and for $B$ any invertible matrix. Then the Kronecker-product of $B$ with itself is invertible with inverse $B^{-1} \otimes B^{-1}$ (can be proven by matrix multiplication), but $A \otimes B$ is zero, and therefore not invertible.

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