A Mnemonic for the Invertible Matrix Theorem

November 10, 2019

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A is an	I	nvertible Matrix.
A is	R	ow equivalent to the n × n identity matrix.
A has n	P	ivot positions.
$A\vec{x} = \vec{0}$ has only the	T	rivial solution.
The columns of A form a	L	inearly independent set.
The transformation x⃗ ↦ Ax⃗ is	O	ne-to-one.
Ax⃗ = b⃗ has at least	A	t least one solution for b⃗ in ℝⁿ.
The columns of A	S	pan ℝⁿ.
The transformation x⃗ ↦ Ax⃗	M	aps ℝⁿ onto ℝⁿ.
There is an m × n matrix	C	such that CA = I.
There is an m × n matrix	D	such that AD = I.
The	T	ranspose of A is invertible.

(Definition of IMT taken from Linear Algebra and its Applications, Fifth Edition by Lay, Lay, and McDonald.)