If 0 is an eigenvalue of A then nullityA 0 True or False ex

If 0 is an eigenvalue of A, then nullity(A) > 0. True or False, explain.

-True. Since 0 is an eigenvalue, A is one-to-one, and thus nullity(A) > 0.

-True. Since 0 is an eigenvalue, there exists a nonzero vector x such that Ax = 0, and thus nullity(A) > 0.

-False. Since 0 is an eigenvalue, there exists a nonzero vector x such that Ax = 0, and thus nullity(A) = 0.

-False. Consider

-False. Consider

	0	1
1	0

Solution

TRUE. Since 0 is an eigenvalue, there exists a nonzero vector x such that Ax = 0, and thus nullity(A) > 0.

If 0 is an eigenvalue, the the deternimant of the matrix becomes zero. So that the system of equations AX=O has

non trivial solution.