Check if a matrix is invertible

Last updated Feb 26, 2022 Edit Source

• Math linear

A [notes/Matrixes|matrix] of size $n \times n$ is invertable if and only if one of the conditions below hold:

1. $A$ is equivalent to $I_n$
2. $A$ has $n$ pivot positions
3. The equation $Ax = 0$ only has the Trivial Solution
4. The columns of $A$ are linearly independent
5. The equation $Ax = b$ has at least one solution for each $b$ in $\mathbb{R}^n$
6. The column of $A$ span $R^n$
7. $A^T$ is invertible as well
8. The columns of $A$ form a basis for $\mathbb{R}^n$
9. $Col A = \mathbb{R}^n$ (Column Space)
10. $Dim(Col A) = n$ (Dimension)