Question

What are the implications of matrix invertibility?

Answer 1

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).