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I don't understand why only a matrix full of zero has a rank = 0.

"the rank of a matrix A is the number of linearly independent rows or columns of A"

If I have a 3x3 matrix

A = [ 1 1 1

1 1 1

1 1 1 ]

assuming a_i denotes the column or row vector i of A. I can say

a_1 = 1*a_2 + 0*a_3 so a_1 is not linearly independant

a_2 = 1*a_1 + 0*a_3 so a_2 is not linearly independant

a_3 = 1*a_1 + 0*a_2 so a_3 is not linearly independant

So why rank A = 1 and not 0 ?

I know i'm missing something, I don't know what!