For what values of $a,b,c$ does a left and/or right inverse exist for $A=\begin{bmatrix} 1 & a \\ 2 & b \\ 3 & c \end{bmatrix}$ exist?

We know that a left inverse matrix $X$ exists such that $XA=I_2$ where $I_2$ is the $2\times 2$ identity matrix so $X$ is a $2\times 3$ matrix. What do we do next? Thanks.

  • $\begingroup$ How do you know that a left inverse exists? What is that left inverse if $\ a=1\ $, $\ b=2\ $ and $\ c=3\ $? $\endgroup$ – lonza leggiera Apr 5 at 0:32
  • $\begingroup$ Hint: if the equation $xA=b$ has a solution, what is the relationship between $A$ and $b$? $\endgroup$ – amd Apr 5 at 0:35
  • $\begingroup$ The fact that "a matrix has left Inverse iff its linear map is an injection and has right inverse iff its linear map is a surjection" may help. $\endgroup$ – Kamyar Mirzavaziri Apr 5 at 0:43
  • $\begingroup$ what have you tried? $\endgroup$ – Manuel Pena Apr 6 at 0:07

Here we can view $A$ as a map from $\mathcal{R}^2$ to $\mathcal{R}^3$.

Having a left inverse is impossible if $A$ is not an one to one mapping.

Image of the mapping $A$ is the column space of $A$.

If the columns space is 1-dimentional then $A$ is not one to one (because by rank-nullity theorem, kernel of $A$ is no trivial).

So to have a left inverse we need $[a,b,c]$ not a multiple of $[1,2,3]$ to guaranty that image of $A$ is not 1-dimentional.

Having a right inverse is impossible as there are no mapping $X$ that is one to one from $\mathcal{R}^3$ to $\mathcal{R}^2$.

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  • $\begingroup$ So would that be the opposite if $A=\begin{bmatrix} 1 & 2 & 3\\ a & b & c \end{bmatrix}$ meaning a left inverse is impossible in this case? $\endgroup$ – squenshl Apr 5 at 1:38
  • $\begingroup$ In this case, $A$ could have right inverse but can't have left inverse and rank of row space is the same as rank of column space. so making $[a b c]$ not a multiple of $[1 2 3]$ would work. $\endgroup$ – user 6663629 Apr 5 at 1:45

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