# What are the implications of matrix invertibility?

Feb 6, 2017

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) $m a t h b f x = m a t h b f 0$ is the only solution to $A m a t h b f x = m a t h b f 0$,

• (3) $m a t h b f x = {A}^{- 1} m a t h b f b$ is the only solution to $A m a t h b f x = m a t h b f b$, 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).