# How would changing of elements of matrix(from zeros to non-zeros) affect its rank.

For a block diagonal matrix of the form $\left[\begin{array}{ccc}A_1 & & O\\ & \ddots & \\ O & & A_L\end{array}\right]$, where $A_1,\ldots,A_L$ are matrices of size $K\times K$, the rank of it is $R\leq KL$. My question is, if I randomly pick up one zero from the off-block-diagonal part and change that to a non-zero element, how would that affect the rank of this matrix (decrease, increase, non-increase, non-decrease, equal)? Intuitively it looks like non-decreasing, but how to give a mathematical proof on that? Thanks.

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$\operatorname{rank}(A+B) \le \operatorname{rank}(A) + \operatorname{rank}(B)$ – user2468 Mar 7 '12 at 2:00

If you change multiple entries off the block diagonal, and change entries that are not in the same $K \times K$ block, all bets are off.