Invertible Matrix Theorem

Assumptions for A, a square n × n invertible matrix

Assumptions for A, a square n × n invertible matrix

Invertible Matrix Theorem

A is row equivalent to

the n × n identity matrix.

A has n amount of ___

pivot positions

The equation Ax = 0 has ____ solution.

only the trivial

(v) The columns of A form a ____ set.

linearly independent

The linear transformation x ↦ Ax

is one-to-one.

The equation Ax = b has at least

one solution for each b in Rn

What can be said about the span of A?

The columns of A span Rn.

The linear transformation x ↦ Ax maps

Rn onto Rn.

What other matrices exist?

There is an n × n matrix C such that CA = I.

There is an n × n matrix D such that AD = I.

What can be said about the transpose of A?

A^T is an invertible matrix.