What are the implications of matrix invertibility?
See below for rough outline.
If an nxn matrix is invertible, then the big-picture 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 non-zero,
#mathbf x = mathbf 0#is the only solution to #A mathbf x = mathbf 0 #,
#mathbf x = A^(-1) mathbf b#is the only solution to #A mathbf x = mathbf b #, and
(4) it's eigenvalues are non-zero.
A singular (non-invertible) 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).