What the rank of a matrix with the elements of one column to be infinity?

Question

Suppose the $m\times m$ real matrix $A$ is positive definite, it is without doubt that $\mathrm{rank} (A) = m$. Now, if all the elements in the first column and first row are assigned to be infinity, what is the rank of $A$?

Numerically, since other elements can be viewed as zero, thus we have $\mathrm{rank}(A) = 2$. However, I wonder whether it is theoretically established.

Thanks for 5xum, the restriction of "real matrix" is removed.

Answer 1

The following two statements are in direct contradiction:

Suppose the $m\times m$ real matrix $A$

if all the elements in the first column and first row are assigned to be infinity

Infinity is not a real number, this, if the all elements are asigned to be "infinity", then the matrix is no longer a real matrix.

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In general:

• The rank of a matrix is defined as the number of linearly independent columns of $A$ (or, equivalently, as the dimension of the image of $A$).
• Linear dependency (and dimensionality) are defined for vector spaces.
• Vector spaces are defined over fields.

The above means that, in order to speak about the rank of a matrix, the elements of that matrix must, by definition, be elements of the field over which the vector space you are examining is defined. So before asking about the rank of a matrix that includes $\infty$ as its element, you need to define which field you are looking at this in. And no, that field cannot be $\mathbb R$, because $\infty\notin\mathbb R$.