invertible matrix

let $\text{[math]}$ be a matrix. if there is a matrix $\text{[math]}$ such that $\text{[math]}$ , then $\text{[math]}$ is a left inverse of $\text{[math]}$ .
[cite:;taken from @calc_hubbard_2015 definition 1.2.11]

let $\text{[math]}$ be a matrix. if there is a matrix $\text{[math]}$ such that $\text{[math]}$ , then $\text{[math]}$ is a right inverse of $\text{[math]}$ .
[cite:;taken from @calc_hubbard_2015 definition 1.2.11]

an invertible matrix is a matrix that has both a left inverse and a right inverse.
[cite:;taken from @calc_hubbard_2015 definition 1.2.13 invertible matrix]

if a matrix $\text{[math]}$ has both a left and a right inverse (is invertible), then it has only one left inverse and one right inverse, and they are identical; such a matrix is called the inverse of $\text{[math]}$ and is denoted $\text{[math]}$ .
[cite:;taken from @calc_hubbard_2015 proposition and definition 1.2.14 matrix inverse]

[cite:;taken from @calc_hubbard_2015 chapter 1 vectors, matrices and derivatives]

if $\text{[math]}$ is an invertible matrix, then $\text{[math]}$ because
$\text{[math]}$ [cite:;taken from @algebra_axler_2024 definition 3.80]

if $\text{[math]}$ and $\text{[math]}$ are invertible square matrices of the same size, then $\text{[math]}$ is invertible and $\text{[math]}$ because
$\text{[math]}$ and similarly $\text{[math]}$ .
[cite:;taken from @algebra_axler_2024 definition 3.80]

for any invertible square matrices $\text{[math]}$ , the inverse of their product is given by:
$\text{[math]}$

$\text{[math]}$

if $\text{[math]}$ is an invertible matrix then $\text{[math]}$ .

$\text{[math]}$

if $\text{[math]}$ is an invertible transformation from the vector space $\text{[math]}$ onto the vector space $\text{[math]}$ then $\text{[math]}$ .

a square matrix $\text{[math]}$ is invertible iff its determinant is non-zero ( $\text{[math]}$ ). conversely, $\text{[math]}$ is not invertible (singular) if and only if its determinant is zero ( $\text{[math]}$ ).

let $\text{[math]}$ . the following are equivalent:

$\text{[math]}$ is invertible.
$\text{[math]}$ .
the columns (or rows) of $\text{[math]}$ are linearly independent (form a basis for $\text{[math]}$ ).
$\text{[math]}$ .
0 is not an eigenvalue of $\text{[math]}$ . (i.e. $\text{[math]}$ ).
the constant term of $\text{[math]}$ is non-zero. (i.e., $\text{[math]}$ ). (if $\text{[math]}$ , then $\text{[math]}$ . if $\text{[math]}$ , then $\text{[math]}$ is a factor of $\text{[math]}$ , so 0 is a root of $\text{[math]}$ , meaning 0 is an eigenvalue).