When checking if a matrix A of size 3x2 can have a left inverse, is this correct:

XA = I

If A is 3x2 then A has a rank of 2. Also, X must be 2x3, which means matrix X has a rank of (2 or 3)?

Matrix I will be a 2x2 identity matrix because X.A is 2x3 * 3x2 = 2x2. Matrix I will have a rank of 2.

Because matrix I's rank does not exceed the rank of X or A, then matrix A can have a left inverse.

My question is: does matrix X have a rank of 2 or 3? It's dimensions are 2x3.

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    $\begingroup$ No $2\times 3$ matrix can ever have rank$~3$ (nor $4$, for the same reason). $\endgroup$ – Marc van Leeuwen Feb 27 '15 at 19:50

If $XA = I$ and $A$ is $3\times2$, then $A$ has a rank of $1$ or $2$. $X$ must be $2\times3$ which again means $X$ could have a rank of $1$ or $2$ (never $3$). And matrix $I$ will be the $2\times2$ identity matrix as you stated.

Now carryout the multiplication in $XA = I$. The element $I(1,1) = 1$ is the dot product of the first row of $X$ by the first column of $A$. Similarly $I(2,1) = 0$ is the dot product of the second row of $X$ by the first column of $A$.

Now consider a case where $X$ has a rank of $1$. If so, the rows of $X$ will be linearly dependent. In this case, if the dot product of one of the rows by any vector were to be equal to zero, the dot product of the other row would also be equal to zero (since the rows are linearly dependent) and it would not be possible for $XA$ to be equal to the identity matrix.

The same is true if $A$ were to have a rank of $1$. Therefore, we can conclude that a necessary condition for $A$ to have a left inverse would be that $A$ be of maximal rank.


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