anonymous
  • anonymous
if Null(A^T) != {0}, then A is not invertible? Where A is an mxn matrix. (!= means not equal to)
Linear Algebra
chestercat
  • chestercat
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math&ing001
  • math&ing001
If A is a mxn matrix and m != n then it can't be invertible. Only square matrices are. It can only have a right or a left inverse.
math&ing001
  • math&ing001
Btw what's the T there in Null(A^T) ?
anonymous
  • anonymous
A^T means Transpose of A

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beginnersmind
  • beginnersmind
"if Null(A^T) != {0}, then A is not invertible?" Right. Let's say A is an nxn matrix so A^T is nxn as well. Since A^T has non-zero vectors in its nullspace it can't be full rank. Remember rankB = n - dim(N(B)). Since A and A^T have the same rank A isn't full rank either. (rankA < n). Therefore it's not invertible. The last step can be understood through the rank formula as well. Since: r < n r = n - dim (N(A)) so dim (N(A)) > 0, therefore Ax = 0 for some nonzero vector and A is not invertible.
anonymous
  • anonymous
for invertible the given matrix must be square so it is not invertible

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