Nullspace of a square matrix A and A^2 are related?

[brownman]

Homework Statement

Say that A is a square matrix. Show that the following statements are true, or give a counter example:
a) If x is in the nullspace of A, then x is in the nullspace of A²
b) If x is in the nullspace of A², the x is in the nullspace of A.

Homework Equations

The Attempt at a Solution

I solved part a, or maybe I didn't. I said

"Ax=0 is our assumption.
A²x = A*Ax = A(0) = 0
so statement a is true."

However, for part b, I stated:

"A²x=0 is our assumption.
Let B=A², so Bx=0 is true.
A*Ax = 0

We have no way of knowing if Ax is true yet.
However if we left multiply by the inverse of A,
we can see that Ax=0. Therefore the statement
b is true unless the determinant of A is zero,
and the inverse does not exist."

However, when trying any and all matrices, some with and some without a determinant equal to zero, and finding the nullspace of the matrix squared and checking it with the original matrix, it always returns a matrix of zero. Ideas? Thanks in advance.

 

[Mark44]

Homework Statement

Say that A is a square matrix. Show that the following statements are true, or give a counter example:
a) If x is in the nullspace of A, then x is in the nullspace of A²
b) If x is in the nullspace of A², the x is in the nullspace of A.

Homework Equations

The Attempt at a Solution

I solved part a, or maybe I didn't. I said

"Ax=0 is our assumption.
A²x = A*Ax = A(0) = 0
so statement a is true."

However, for part b, I stated:

"A²x=0 is our assumption.
Let B=A², so Bx=0 is true.
A*Ax = 0

We have no way of knowing if Ax is true yet.
However if we left multiply by the inverse of A,
we can see that Ax=0. Therefore the statement
b is true unless the determinant of A is zero,
and the inverse does not exist."

However, when trying any and all matrices, some with and some without a determinant equal to zero, and finding the nullspace of the matrix squared and checking it with the original matrix, it always returns a matrix of zero. Ideas? Thanks in advance.

Part a looks fine.
Part b - see if you can find a matrix A (2 x 2 is fine) such that A² = 0, even though A ≠ 0.

 

[brownman]

Oh... Okay I get it now.

If I use the matrix

0 0
0 1 = A and

0 0
0 0 = A²

The nullspace of A² has infinite solutions
and the nullspace of A will have at least one x value
that will have to be zero in order for it to be a valid
equation, so the A² nullspace can not
transfer over.

Thanks :)