# question from my exam: trivial solution only or more solutions?

$A$ is an $n\times n$ matrix, $n \ge 3$. Assume $A^2 = 0$. Which one is true and which one is false and explain:

1. The linear system $Ax = 0$ has only trivial solution.

2. The linear system $Ax = 0$ has more than only the trivial solution.

3. The linear system $(I - A)x$ has only trivial solution.

• Yes you understood correctly. Can you tell me why is it equivalent to A = 0 ? And about 3. I proved it by muliplying in I+A snd got I , is it a good explanation to show there's only trivial solution? Jul 9, 2013 at 23:47
• Sorry, I meant A^2 = 0, very sorry Jul 9, 2013 at 23:54
• What's the correct answer then? Sorry. .. Jul 9, 2013 at 23:55
• Yes, multiplying by $I+A$ is the way for 3. Jul 9, 2013 at 23:57
• If $Ax=0$ has only the trivial solution, then for $x\ne0$, $Ax\ne0$. Then $A(Ax)=0$, a contradiction. So 1. is false. Item 2. is just the negation of item 1.. Jul 10, 2013 at 0:06

Hint: $$0=\det(A^2)=\det(A)^2$$ What does that tell you about the rank of $A$? is $A$ invertable?
• @user84636 - you can't multiply $x$ by $A$ on the right.. Jul 10, 2013 at 5:54