# Rank of a degenerate conic

This question comes from projective geometry. A degenerate conic $C$ is defined as $$C=lm^T+ml^T,$$

where $l$ and $m$ are different lines. It can be easily shown, that all points on $l$ and m lie on the $C$. Because, for example, if $x\in l$, then by definition $l^Tx=0$ and plugging it into conic equation makes it true.

Question: Find the rank of $C$.

(We can limit ourself to 3-dimensional projective space.)

P.S. I'm reading a book, where it is guessed and checked, but I would like to have a proof without guessing. I do not provide the guess, since it can distract you, but if you really need it just leave a note.

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The rank of $lm^T$ is one. The same goes for $ml^T$. In most cases, the rank of the symmetric matrix $C$ as you define it will be 2. This corresponds to a conic degenerating into two distinct lines. If the lines $l$ and $m$ should coincide, though, the rank of $C$ will be 1.