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Let $A$ and $B$ be matrices of $M_{n}(K)$. Show that if $AB$ is invertible the matrices $A$ and $B$ are invertible.

So i know how to find the inverse of a matrice, i know how to do the product of two matrices and i know the definition of invertibility but somehow i can't find a way to prove it.

Any help to point me in the right direction would help me alot.

Thank you.

spexel
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3 Answers3

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Let $C$ be the inverse of $AB$, $(AB)C =I_n$ implies $A(BC) = I_n$, thus the linear morphism $L(A)$ defined by $A$ on $k^n$ is surjective since the dimension of $k^n$ is finite, $dim(Ker(L(A))+dim(Im(L(A)) =n$, $dim(Im(L(A)) =n$ implies $ker(L(A)) = 0$ and $A$ inversible, $B= A^{-1}C^{-1}$ is inversible.

Tsemo Aristide
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  • I still don't get it – spexel Oct 17 '15 at 16:32
  • I have this in my book if $AB^{-1}=A^{-1}B^{-1}$ then we know $A$ and $B$ are invertible.

    Demonstration:

    We have $(AB)(B^{-1})A^{-1}=A(BB^{-1})A^{-1}=A(I_{n})A^{-1}=AA^{-1}=I_{n}$ and $(B^{-1}A^{-1})(AB)=B^{-1}(A^{-1}A)B=B^{-1}I_{n}B=B^{-1}B=I_{n}$

    am I supposed to use that to solve for this problem?

     – spexel Oct 17 '15 at 16:48
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Hint:

Use the determinant criterion for invertibility.

Bernard
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Hint 1: The rank of a product is at most equal to the rank of the factors, so the rank of $A$ and $B$ is…

Hint 2: A square matrix is invertible if and only if it has either a left or a right inverse.

egreg
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