If $M, P$ are Nonsingular, then Exists a Matrix $N$ such that $MN=P$

Nonsingular matrix and singular matrix problems and solutions

Problem 657

Suppose that $M, P$ are two $n \times n$ non-singular matrix. Prove that there is a matrix $N$ such that $MN = P$.

 
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Proof.

As non-singularity and invertibility are equivalent, we know that $M$ has the inverse matrix $M^{-1}$.

Let us think backwards. Suppose that we have $MN=P$ for some matrix $N$, which we want to find.
Then multiply $M^{-1}$ on the left of the equation $MN = P$ yields $N = M^{-1} P$.


This is the matrix we are looking for, as $ M N = M ( M^{-1} P) = P$.


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