What are the implications of matrix invertibility?
1 Answer
Feb 6, 2017
See below for rough outline.
Explanation:
If an nxn matrix is invertible, then the bigpicture consequence is that its column and row vectors are linearly independent.
It is also (always) true to say that if an nxn matrix is invertible:

(1) its determinant is nonzero,

(2)
#mathbf x = mathbf 0# is the only solution to#A mathbf x = mathbf 0 # , 
(3)
#mathbf x = A^(1) mathbf b# is the only solution to#A mathbf x = mathbf b # , and 
(4) it's eigenvalues are nonzero.
A singular (noninvertible) matrix has at last one zero eigenvalue. But there is no guarantee that an invertible matrix can be diagonalised or vice versa.
Diagonalisation will only happen when a matrix delivers up a full set of eigenvectors (which can occur where an eigenvalue is zero).