What are the implications of matrix invertibility?

1 Answer
Feb 6, 2017

Answer:

See below for rough outline.

Explanation:

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,

  • (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 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).