Matrix Equivalency Statements

Theorem

Let $A$ : $n\times n$ square matrix

Then the following statements are equivalent:

• (a) $A$ is invertible
• (b) $A\mathbf{x}=0$ 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) $A\mathbf{x}=\mathbf{b}$ is consistent for every $n\times 1$ matrix $\mathbf{b}$
• (f) $A\mathbf{x}=\mathbf{b}$ has exactly one solution for every $n\times 1$ matrix $\mathbf{b}$
• (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 $A$ span $R^n$
• (k) The row vectors of $A$ span $R^n$
• (l) The column vectors of $A$ form a basis for $R^n$
• (m) The row vectors of $A$ for a basis for $R^n$
• (n) $A$ has rank $n$
• (o) $A$ has nullity $0$
• (p) The orthogonal complement of the null space of $A$ is $R^n$
• (q) The orthogonal complement of the row space of $A$ is $\{0\}$