Question

Let A be a 2×2 matrix, and assume that A2 = 0.
Select ALL statements below which must be true.
• A. A = 0
• B. (I+A)(I−A) = I
• C. There is a column X such that X not= 0 and AX = 0
• D. If AX = 0, then X = 0
• E. None of the above

Answer 1

hellooooooo no one is here !!!!!

Answer 2

Do you have any guesses or ideas to which might be true or not true?

Answer 3

sure B and C

Answer 4

C is definitely true. good job on that one. However, I dont believe B is true. Let me see if I can come up with a counter example.

Answer 5

OH, nvm nvm, you are right. For some reason I was thinking (I+A)^2

Answer 6

yes, B is true as well.

Answer 7

\[\left[\begin{matrix}-1 &1 \\ -1 & 1\end{matrix}\right] \times \left(\begin{matrix}t \\ t\end{matrix}\right)=\left(\begin{matrix}0 \\0\end{matrix}\right)\]

Answer 8

what about D

Answer 9

I can come up with a counter example to D for sure. One sec.

Answer 10

Basically, since c is true, that automatically implies that D is false.

Answer 11

however \[\left[\begin{matrix}0 & 1 \\ 0 & 0\end{matrix}\right] \times \left(\begin{matrix}0 \\ 0\end{matrix}\right)=\left(\begin{matrix}0 \\0\end{matrix}\right)\] so X can be zero

Answer 12

yeah, the zero vector will always be a solution to the equation:\[Ax=0\]its what is called the trivial solution. However, given that A^2 = 0, if we have to solve the matrix equation Ax = 0 , we cannot conclude that x is the zero vector. This is because there are other non-zero vectors that are also solutions to Ax = 0.

Answer 13

D is saying "If Ax = 0, then x must be zero", but we know that isnt true, since C is true.

Answer 14

thanks a lot :)