Keywords: invertible
$n \times n$ matrix invertible properties:
Let $A$ be a square $n \times n$ matrix. Then the following statements are equivalent. That is, for a given $A$, the statements are either all true or all false.
a. $A$ is an invertible matrix.
b. $A$ is row equivalent to the $n \times n$ identity matrix.
c. A has $n$ pivot positions.
d. The equation $Ax = 0$ has only the trivial solution.
e. The columns of $A$ form a linearly independent set.
f. The linear transformation $x \rightarrow Ax$ is one-to-one.
g. The equation $Ax = b$ has at least one solution for each $b$ in $R^n$.
h. The columns of $A$ span $R^n$.
i. The linear transformation $x \rightarrow Ax$ maps $R^n$ onto $R^n$.
j. There is an $n \times n$ matrix $C$ such that $CA = I$.
k. There is an $n \times n$ matrix $D$ such that $AD = I$.
l. $A^T$ is an invertible matrix.
m. The columns of $A$ form a basis of $R^n$:
n. $Col A = R^n$
o. $dim Col A = n$
p. $rank A = n$
q. $Nul A = {0}$
r. $dim Nul A = 0$
s. The number $0$ is not an eigenvalue of $A$.
t. The determinant of $A$ is not zero.
u. $(Col A)^\perp = {0}$.
v. $(Null A)^\perp = R^n$.
w. $RowA = R^n$.
x. $A$ has $n$ nonzero singular values.
