# Kronecker product of identity and matrix product

How is the following property true? Let $$I$$ be the identity matrix and $$A$$, $$B$$ be appropriately sized real matrices. Then $$I \otimes \left(\left( I \otimes A\right) B \right) = \left( I \otimes I \otimes A \right)\left( I \otimes B\right).$$ I think I'm missing something basic about the Kronecker product.

The kronecker product is associative, so $$(A\otimes B)\otimes C=A\otimes(B\otimes C)$$ and use that $$(AB)\otimes(CD)=(A\otimes C)(B\otimes D)$$. Now, you can just expand.
• Yes. I didn't think that $I = I^2$. Thanks! – G. Gare Jan 28 at 8:37