Matrix - Invertible

Invertible Matrices and Matrix Inverse

The square matrix $A$ is invertible if there is a matrix $A^{-1}$ such that:

$$A\cdot A^{-1}=A^{-1}\cdot A=I$$

Where: $I$ is the identity matrix.

If $A$ is not invertible, it is called a singular matrix.

Equivalent Statements

The following statements are equivalent for an $n\times n$ matrix $A$:

1. $A$ is invertible
2. $A$ is row equivalent to the identity matrix
3. $A$ has $n$ pivot positions
4. The system $A\times \vec{x}=\vec{0}$ has only the trivial (zero) solution
5. The system $A\times \vec{x}=\vec{b}$ has at least one solution for every $\vec{b}$
6. The vector span of the columns of $A$ span all of ${\mathbb{R}}^n$

Finding the Inverse

Using Matrix Row Reduction

Given $A\in n\times n$ matrix, $A^{-1}$ can be found by constructing a matrix

$$\left[\begin{array}{cc}A& I_n\end{array}\right]$$

and performing row reduction until it is in RREF form. The result will be

$$\left[\begin{array}{cc}{I}_n& A^{-1}\end{array}\right]$$

If $I$ is not found on the left (there isn’t a pivot position in each row), then $A$ is not invertible.

For 2x2 Matrix

Given $A\in 2\times 2$ matrix, the inverse can be found using the formula:

$$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]⟹A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$$

Where $ad-bc\ne 0$ (the singular matrix case).

Properties

Inverse of an Inverse is the Original Matrix

$$(A^{-1}{)}^{-1}=A$$

Inverse of a Transpose is the Transpose of the Inverse

$$(A^T{)}^{-1}=(A^{-1}{)}^T$$

Inverse of a Product is the Product of the Inverses in Reverse Order

$$(AB{)}^{-1}=B^{-1}{A}^{-1}$$