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Correct me if I'm wrong but if a null space of a matrix A is nontrivial would it be correct to say that it is the opposite of the list of points in the Invertible Matrix Theorem?

  • A is an invertible matrix
  • A is row equivalent to the identity matrix
  • A has n pivot columns
  • The equation has only a trivial solution to ax=0
  • The columns of A are linearly independent
  • The equation Ax=b has at least one solution for each b in Rn
  • The column of A span Rn
  • maps Rn onto Rn
  • There is a nxn matrix C such that CA is equal to the identity matrix
  • There is an nxn matrix D such that AD is equal to the identity matrix ....
quid
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MasterExploder
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2 Answers2

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A matrix whose nullspace is non-trivial, that is does not consist only of $0$, is never invertible. And, conversely, every non-invertible matrix has non-trivial null space. (That is if we restrict our discussion to square matrices.)

Thus, asserting a (square) matrix has non-trivial nullspace is equivalent to asserting it is not invertible.

Thus, such a matrix indeed does not have any of the properties you list; and every matrix that does not have the properties you list has non-trivial null space.

In that sense, yes, it is somehow the opposite. Having non-trivial nullspace characterizes a matrix as not invertible, while all the properties you list characterize a matrix as invertible.

quid
  • 42,135
2

I am not sure what you are asking, but an equivalent condition for invertibility is that the null space of $A$ is trivial (i.e., $\operatorname{ker}A=\{0\}$).

parsiad
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