Summing the Matrix

Calculus Level 2

Alice and Bob saw this matrix with infinite rows and columns:

\[ \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & \cdots \\ -1 & 1 & 0 & 0 & 0 & \cdots \\ 0 & -1 & 1 & 0 & 0 & \cdots \\ 0 & 0 & -1 & 1 & 0 & \cdots\\ 0 & 0 & 0 & -1 & 1 & \cdots\\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{pmatrix}. \]

They wanted to sum up all the elements in it.

Alice: "The sum of each column is 0, so the sum of all the numbers must be 0."
Bob: "The sum of all the rows except the first one is 0. The first row adds up to 1. Hence, the sum of all the numbers is 1."

Who is correct?

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