📂Matrix Algebra

Conditions for a Matrix Being Invertible

Theorem¹

Let $A$ be a square matrix of size $n\times n$. Then the following propositions are all equivalent.

(a) $A$ is an invertible matrix.

(b) The homogeneous linear system $A\mathbf{x}=\mathbf{0}$ has only the trivial solution.

(c) The reduced row echelon form of $A$ is $I_{n}$.

(d) $A$ can be expressed as a product of elementary matrices.

(e) $A\mathbf{x}=\mathbf{b}$ has a solution for all $n\times 1$ matrices $\mathbf{b}$.

(f) $A\mathbf{x}=\mathbf{b}$ has exactly one solution for all $n\times 1$ matrices $\mathbf{b}$. That is, $\mathbf{x}=A^{-1}\mathbf{b}$ is satisfied.

(g) $\det (A) \ne 0$

(h) The column vectors of $A$ are linearly independent.

(i) The row vectors of $A$ are linearly independent.

(j) The column vectors of $A$ generate $\mathbb{R}^{n}$.

(k) The row vectors of $A$ generate $\mathbb{R}^{n}$.

(l) The column vectors of $A$ are a basis for $\mathbb{R}^{n}$.

(m) The row vectors of $A$ are a basis for $\mathbb{R}^{n}$.

(n) The rank of $A$ is $n$.

(o) The nullity of $A$ is $0$.

(p) The orthogonal complement of the null space of $A$ is $\mathbb{R}^{n}$.

(q) The orthogonal complement of the row space of $A$ is $\left\{ \mathbf{0} \right\}$.

(r) None of the eigenvalues of $A$ is $0$.

(s) $A^{T}A$ is invertible.

(t) The kernel of $T_{A}$ is $\left\{ \mathbf{0} \right\}$.

(u) The image of $T_{A}$ is $\mathbb{R}^{n}$.

(v) $T_{A}$ is a one-to-one function.

Proof

(a) $\iff$ (b) $\iff$ (c) $\iff$ (d)

(a) $\iff$ (e) $\iff$ (f)