Suppose we have $A^{m*n}$ and $B^{n*m} $ matrices. What constraints has to be satisfied by $A$ and $B$ so that the product of these result in the identity matrix $I$. I fund that it is possible but I cannot find the constraints for $A$ and $B$ which I really need. Thanks for the help.
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The constraints should, probably, be black magic or something like that: the dot or inner product of two vectors in a linear space with inner product gives us a scalar, not another vector. Perhaps you meant simply product? – DonAntonio May 16 '17 at 09:47
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Yes sorry you are right – DalekSupreme May 16 '17 at 09:50
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I suppose by dot product you mean usual matrix product.
So if $$AB=I_m$$ then $B$ is called the right inverse of $A$ and $A$ is the left inverse of $B$.
If $$BA=I_n$$ then $A$ is the right inverse of $B$ and $B$ is the left inverse of $A$.
A matrix has right inverse iff its columns are independent.
A matrix has left inverse iff its rows are independent.
However, a non-square cannot have both left and right inverse.
velut luna
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Too big for a comment, but for a pair of $2\times 3$ and $3\times 2$ matrices we get $$\left(\begin{array}{lll}a&b&c \\ d&e&f \end{array}\right)\left(\begin{array}{ll}g&h\\i&j\\k&l\end{array}\right)=\left(\begin{array}{ll}1&0\\0&1\end{array}\right).$$ This implies $$ag+bi+ck=1\\dg+ei+fk=0\\ah+bj+cl=0\\dh+ej+fl=1.$$ So these are the constraints, which you can adapt to any sized matrices.
pshmath0
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