- #1
brownman
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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.