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**Let A = [a b; c d] a 2x2 matrix with complex entries. Suppose that A is row-reduced and also that a+b+c+d =0 . Prove that there are exactly three such matrices....**

so i realize that there are seven possible 2x2 matrices that are row-reduced.

[1 0; 0 1], [0 1; 1 0], [0 0; 1 0], [0 0;0 1], [1 0; 0 0], [0 1; 0 0], [0 0; 0 0]...

and the only one that satisfies the restriction is the last. What am i missing? thx

This is the problem 6 of the section 1.3 of HOFFMAN and KUNZE - Linear Algebra.