I have been looking around for an answer to this question for a bit. I'm asking about this because it would really help me with a problem I have. Essentially, what I want to know is whether a block matrix of the form
$$A = \begin{pmatrix} I & B \\ 0 & C \end{pmatrix}$$
where $I$ is the identity matrix, $B$ is some arbitrary matrix, $C$ is a non-singular matrix. Is it true that rank$A$ = rank$I$ + rank$C$? I know this is true when $B = 0$ but not sure about something like this. Any help would be greatly appreciated.
Well, for block upper triangular matrices, we have the same result than for regular matrices:
$$ \det(A) = \det(I)\cdot \det(C) = \det(C) \neq 0 $$
so $A$ is invertible and should have full rank. This coincides with the sum of the ranks of $A$ and $C$ because they also have full rank, since that they are invertible.