Question

for partitioned matrix A=[A1 A2 ; 0 A3]
show rank(A)=rank(A1)+rank(A3)

Answer 1

A=\[\left[\begin{matrix}A1 & A2 \\ 0 & A3\end{matrix}\right]\]

Answer 2

i think in this way i dont know exactly;
rank(A)=sum of ranks of diogonal mat

Answer 3

@mathmale

Answer 4

Hi, Divine One,
I've had some experience with partitioned matrices, but that was literally decades ago. Have you tried Internet searches for info on topics such as this one? Occasionally one comes across really good, clear explanations that way.
I'm not saying that the following is exactly relevant to your question, but it just might help:

http://link.springer.com/chapter/10.1007%2F978-3-642-10473-2_6#page-1

You could do searches for "rank of a matrix" and "rank of a partitioned matrix."

Good luck!

PS: Partitioning matrices is a really neat way to reduce the work involved in evaluating the determinants of the partiitions.