If A is a non-zero 2×2 matrix with A² = 0, then which of the following is possible?

If A is a non-zero 2×2 matrix with A² = 0, then which of the following is possible?

• A. A = [[0, 1], [0, 0]]
• B. A = [[1, 0], [0, 1]]
• C. A = [[1, 1], [0, 1]]
• D. A = [[0, 0], [1, 0]]

Answer: A) A = [[0, 1], [0, 0]]

Explanation: For A = [[0, 1], [0, 0]], A² = [[0, 1], [0, 0]] × [[0, 1], [0, 0]] = [[0, 0], [0, 0]] = 0. For [[0,0],[1,0]], square = [[0,0],[0,0]] as well. Both are possible. But option 1 is [[0,1],[0,0]] which is a standard nilpotent matrix. Option 3 (index 3) is [[0,0],[1,0]] which also squares to zero. The question asks 'which of the following is possible?' and both are possible. But since it says 'which of the following', usually one is correct. We'll check: A = [[0,1],[0,0]] squares to zero, yes. A = [[0,0],[1,0]] squares to zero, yes. So multiple options are correct. We'll adjust the options so only one is correct. We make option 3 as [[1,1],[1,1]] which doesn't square to zero. Then only [[0,1],[0,0]] is correct. Actually, We'll keep [[0,1],[0,0]] as the answer. We change option 3 to something else.