I was recently asked to find the right inverse of some matrixes. I found that all three of them were invertible, so it was just a matter of finding their inverses, which would be exactly the same as the right inverses.

What if a matrix is not invertible?

Basically, what I want to know is, if a matrix is not invertible, does it mean that there are no left and/or right inverses at all? That is,

$$|A| = 0 \iff \not \exists B(AB = I \ \lor \ BA = I)$$

  • $\begingroup$ For square matrices, if $AB = I$, then also $BA=I$. Therefore, if a singular square matrix had a right inverse, that would contradict its singularity. $\endgroup$ – Rustyn Apr 6 '14 at 7:45

If your matrix $A$ is square, then $A$ has a left inverse if and only if $A$ is invertible. Also, if $A$ is square, then $A$ has a right inverse if and only if $A$ is invertible.

It is possible to construct noninvertible nonsquare matrices with a right inverse. For example, consider the projection $\Bbb R^2\to \Bbb R$. This is given by the matrix $$\begin{bmatrix}1 &0\end{bmatrix}$$ Its right inverse is $$\begin{bmatrix}1\\ 0\end{bmatrix}$$ Of course, the second matrix in this example is an example of a noninvertible matrix with a left inverse.

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    $\begingroup$ So, for a square matrix, if it isn't invertible, then there is no such thing as left/right inverses? $\endgroup$ – Zol Tun Kul Apr 6 '14 at 7:51
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    $\begingroup$ @Omega That is correct. $\endgroup$ – Brian Fitzpatrick Apr 6 '14 at 7:52

If $$AB=I_n$$ and since $I_n$ is bijective then it's a well known result that $A$ is surjective and $B$ is injective but this is equivalent to say that $A$ and $B$ are bijective in finite dimenesional space due to the rank-nullity theorem. Hence right or left inverse is equivalent to the inverse matrix.

  • $\begingroup$ You are assuming that the matrices are square. While OP might be assuming that as well, it is not said in the question, so I think you should qualify your answer by mentioning this additional hypothesis. $\endgroup$ – Marc van Leeuwen Apr 6 '14 at 10:52

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