Probably an easy question, but can't find anything about it on the internet and it's not in my book. I know the product of two inverse matrices is an inverse matrix. Is it possible to get an inverse matrix if you multiply an inverse matrix with a non invertible matrix? Same with two non-invertible matrices.

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    $\begingroup$ No, since $\text{det}(AB) = \text{det}(A)\text{det}(B)$. $\endgroup$ – MathematicsStudent1122 Dec 18 '17 at 15:19
  • $\begingroup$ Presumably, you mean square matrices. For square matrices, how does the determinant relate to invertibility? $\endgroup$ – quasi Dec 18 '17 at 15:19
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    $\begingroup$ Thanks, the determinant of a non invertible matrix is 0 so because of det(AB)=det(A)det(B) => det(AB)=0 $\endgroup$ – holodorum Dec 18 '17 at 15:22

In case you haven't seen determinants suppose that $AB$ is invertible. Let $C$ be the inverse of $AB$ then:

\begin{align} (AB)C = I \\ A(BC) = I \end{align}

So $A$ is invertible.


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