# Let A be the n × n matrix whose i, j entry is i + j for all i, j = 1, 2, . . . , n. What is the rank of A? [closed]

I tried finding the solution by assuming i and j are both n, but I'm not sure if this is the proper direction to go.

## closed as off-topic by Leucippus, Shailesh, Cesareo, Lee David Chung Lin, SaadMar 15 at 0:48

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• It looks like you may have a fundamental misunderstanding of what the matrix looks like. Try writing out the matrix for, say, a $3 \times 3$ example and see if that helps you understand what's going on. – Robert Shore Mar 14 at 23:18

Hint: the first column is $$\mathbf v=(2,3,\dots,n)^T$$. Furthermore, the $$i$$th column is given by $$\mathbf v+(i-1)\mathbf 1$$ where $$\mathbf 1\in\mathbb R^n$$ is the vector whose entries are all $$1$$s. What does this tell you about the number of column vectors of the matrix that are linearly independent?
• Not that it changes the argument at all, but the entries in the first column are $2,3,....,n,n+1$. – Ned Mar 14 at 23:35